IN MEMORIAM 
FLORiAN CAJORl 





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EATON'S 



ELEMENTARY ALGEBRA 



DESIGNED FOR THE 



USE OF HIGH SCnOOLS AND ACADEHES. 



WILLIAM F. BRADBURY, A. M., 

■OPKIXS MASTER Vf THE CAMBRIDGE niGH SCHOOL ; AOTUOR OF A TREATI8K OK 

TsmoNOiurKr Aia> soanrufG, and of an kuementart oecxetbt. 



BOSTON: 
THOMPSON, BIOELOW, AND BROWN 

26 & 29 CORMBILU 



EATON AND BRADBURY'S 

used with unexampled success in the best schools and 
academies of the country. 

Eaton's Primary Arithmetic 
Eaton's Intellectual Arithmetic. 
Eaton's Common School Arithmetic. 
"•: .•'•.EatoSj's" ffi^fa* School Arithmetic. 

►V. '» './* ^iTok's'.ELi^^i^Ta £)p Arithmetic. 

Eaton's Grammar School Arithmetic. 

Bradbury's Eaton's Elementary Algebra. 
Bradbury's Elementary Geometry. 
Bradbury's Elementary Trigonometry. 
Bradbury's Geometry and Trigonometry, in one volnme. 
Bradbury's Trigonometry and Surveying. 

Keys of Solutions to Common School and High 
School Arithmetics, to Elementary Algebra, Geom- 
etry, AND Trigonometry, and Trigonometry and Sur- 
veying, for the use of Teachers. 



Entered according to Act of Congress, in the year 1868, 

BY WILLIAM F. BRADBURY AND JAMES H. EATON, 

in the Clerk's Office of the District Court of the District of Massachusetts. 



University Press : Welch, Bigelow, & Co., 

Cambridge. 



CAJORl 



'^ 



PREFACE 



It was the intention of the author of Eaton*s Arith- 
metics to add to the series an Algebra, and he had com- 
menced the preparation of such a work. Although its 
completion has devolved upon another, the author, as far 
as practicable in a work of this character, has followed 
the same general plan that has made the Arithmetics so 
popular, and spared no labor to adapt the book to the 
wants of pupils commencing this branch of mathematics. 

A few problems have been introduced in Section II., to 
awaken the pupiPs interest in Algebraic operations, and 
thus prepare him for the more abstract principles which 
must be mastered before the more difficult problems can be 
solved. Special attention is invited to the arrangement 
of the equations in Elimination ; to the Second Method of 
Completing the Square in Affected Quadratics ; and to the 
number and variety of the examples given in the body 
of the work and in the closing section. 

The Theory of Equations, the Explanation of Negative 
Results, of Zero and Infinity, and of Imaginary Quantities, 
are omitted, as topics not appropriate to an Elementary 
Algebra. It may also be better for the younger pupils to 



Qi 1 ?59 7 



IV PREFACE. 

pass over the two theorems in Art. 74, until they become 
more familiar with algebraic reasoning. 

While the book has not been made simple by avoiding 
the legitimate use of the negative sign before a parenthesis 
or a fraction, the difficulty which is caused to beginners 
by the introduction of negative indices in simple division 
has been obviated by deferring their introduction to the 
section on Powers and Roots, where they are fully ex- 
plained. 

The utmost conciseness consistent w^ith perspicuity has 
been studied throughout the work. It is hoped the book 
will commend itself to both teachers and pupils. 

W. F. B. 

Cambbidge, Mass., May 17, 1888. 



CONTENTS. 



SECTION I. 

Taom 

Defixitioxs A!n> Notation . 7 

The SigoA 7 I Axioms 9 

SECTION II. 
Alqeobaic OpzKAnoxs 10 

SECTION III. 
Dfn5Tno:fs asi> Notahon (Contioaed from Section I.) 14 

SECTION IV. 
Additiox 19 

SECTION V. 
SCBTRACnOM 25 

SECTION VI. 
MoiTirucATiox 81 

SECTION VII. 
Dnnsiox 87 

SECTION VIII. 
Dbxoxstkatiox or Tokobzms 43 

SECTION IX. 
Factorixo 47 

SECTION X. 
OuATBST Comiox DirisoR 54 



LXAST COXXOK MOLTtPlI 



SECTION XI. 



Pa.^cTioxs 



8BCTI0N XII. 



fl.«n«ral Prinriple* 6.3 To inultiply a Fraction by an Integral 

Picrn« of Fr.irtions 04 QimnMty 78 

To n'liKc a Frirtion toit^lnwpst tprm« 05 To multiply an Int«^n^l Quantity by 

To re lure FrartionB to equivalent Frac- a Kmrtlon . 75 

tioti* hiving a ('omnion Denominator C* To <livi le a Fraction by an Integral 

To all Fnetlorw 68 | Quantity 76 

To subtract Fmotions 09 To dirlde an Integral Quantity by a 

To reduce a Mixed Quantity to an Im- j Fraction 77 

proper Fraction 70 To multiply a Fraction by a Fraction . 77 

To rwluce an Improper Fraction to an To divide a Fraction by a Fraction . . 79 

Integral or Mixed Quant'ty ... 72 



VI CONTENTS. 

SECTION XIII. 
Equations op thk First Degree containing but one Unknoww Quaittitt . . • 82 

Definitions 82 ' Reduction of Equations 86 

Transposition 83 Problems 91 

Clearing of Fractions 85 1 

SECTION XIV. 

Equations of the First Degree containing two Unknown Quantities . . . , 104 

Elimination by Substitution .... 105 I Elimination by Combination .... 108 

Elimination by Comparison .... 106 | Problems • . 112 

SECTION XV. 
Equations of the First Degree containing more than two Unknown Quan- 



118 



SECTION XVI. 
Powers and Roots 125 



Negative Indices 125 

Multiplication and Division of Powers . 
of Monomials 126 

Transferring Factors from Numerator 
to Denominator, or Denominator to 
Numerator of a Fraction .... 127 

Involution of Monomials 129 



Involution of Fractions ...... 130 

Involution of Binomials 131 

Square Root of Numbers 139 

Cube Root of Numbers 142 

Evolution of Monomials 147 

^-quare Root of Polynomials .... 148 

To find any Root of a Polynomial . . 152 



SECTION XVII. 

Radicals 154 

Definitions 154 To add Radicals 160 

To reduce a Radical to its Simplest Form 154 To subtract Radicals 161 

To reduce a Rational Quantity to the To multiply Radicals 162 

form of a Radical 157 To divide Radicals 163 

To reduce Radicals having different In- To involve Radicals 155 

dices to equivalent ones having a To evolve Radicals 166 

Common Index 158 Polynomials having Radical Terms . 167 

SECTION XVIII. 
Pure Equations which require in their Reduction either Involution or Evo- 



170 



SECTION XIX. 

Affected Quadratic Equations 178 

Completing the Square 178 I Third Method of Completing the Square 185 

Second Method ofCompleting the Square 182 I Problems 193 

SECTION XX. 
Quadratic Equations containing two Unknown Quanttties 196 

SECTION XXI. 
Ratio and Proportion 207 

SECTION XXII. 
Arithmetical Progression 215 

SECTION XXIII. 
Geometrical Progression 225 



SECTION XXIV. 

MlSCXLLAinOUS EZAMFLZS 



ELEMENTARY ALGEBRA. 



SECTION 1. 

DEFINITIONS. 
!• Mathematics is the science of quantity. 

2« Quantity is that which can be measured ; as distance, 
time, weight. 

3* Arithmetic is the science of numbers. In Arithmetic 
quantities are represented by figures. 

4. Algebra is Universal Arithmetic. In Algebra quan- 
tities are represented by either letters or figures, and their 
relations by signs. 

NOTATION. 

5. Addition is denoted by the sign +, called plus ; thus, 
3 + 2, i. e. 3 plus 2, signifies that 2 is to be added to 3. 

6. Subtraction is denoted by the sign — , called minus; 
thus, 7 — 4, i. e. 7 minus 4, signifies that 4 is to be sub- 
tracted from 7. 

7i Multiplication is denoted by the sign X ; thus, 6X5 
signifies that 6 and 5 are to be multiplied together. Be- 
tween a figure and a letter, or between letters, the sign X 
is generally omitted ; thus, Ca6 is the same as 6 X a X 6. 
Multiplication is sometimes denoted by the period ; thus, 
8 . 6 . 4 is the same as 8 X 6 X 4. 



8 ELEMENTARY ALGEBRA. 

8. Division is denoted by the sign -^ ; thus, 9 -r- 3 sig- 
nifies that 9 is to be divided by 3. Division is also indi- 
cated by the fractional form ; thus, f is the same as 9 -i- 3. 

9. Equality is denoted by the sign = ; thus, $1 = 100 
cents, signifies that 1 dollar is equal to 100 cents. An ex- 
pression in which the sign = occurs is called an equa- 
tion, and that portion which precedes the sign = is called 
the Jirsl member, and that which follows, the second mem- 
bef.l .','".' * ' ; , ■ 

10. Inequality is denoted by the sign > or <, the 
smaller quantity always standing at the vertex ; thus, 
8 > 6 or 6 < 8 signifies that 8 is greater than 6. 

11. Three dots .•. are sometimes used, meaning hence, 
therefore. 

12. A Parenthesis ( ), or a Vinculum , indicates 

that all the quantities included, or connected, are to be 
considered as a single quantity, or to be subjected to the 
same operation ; thus, (8 + 4) X 3 = 12 X 3, or = 24 
+ 12 = 36 ; 21 — 6 -=- 3 = 15 -f- 3, or = T — 2 = 5. 
Without t^e parenthesis, these examples would stand 
thus :8 + 4X3 = 8 + 12 = 20;21 — 6-^3 = 21 
— 2 = 19; the sign X. in the former, not affecting 8j 
nor the sign -i-, in the latter, 21. 



Examples. 

1. g + t — 3 + 4 = how many? 

2. (9 + 15) -r- 3 = how many? 

3. — — — X 14 = how many ? 

4. (14 + 13) X (5 — 2) = how many? 

5. 10 + (7 — 4) -^ 3 X 4 = how many ? 



6. 25 — (6 + T) = how many? 
T. 150 -^ (18 — 11) = how many? 



DEFINITIONS. 9 

8. Prove that 175 + 8 — 49 = 14 + 190 — 64 — 16. 

9. Prove that 216 — 44 + 14 > 144 + 13 — T5. 

10. Place the proper sign (=, >, or <) between these 
two expressions, (247 + 104) and (546 — 195). 

11. Place the proper sign (=, >, or <) between these 
two expressions, (119 — 47 + 16) and (317 — 104). 

12. Place the proper sign (=, >, or <) between these 
two expressions, (417 + 31) — (187 — 72) and (127 + 
179). 

AXIOMS. 

IS. All operations in Algebra are based upon certain 
self-evident truths called Axioms, of which the following 
are the most common : — 

1. If equals are added to equals the sums are equal. 

2. If equals are subtracted from equals the remainders 
are equal. 

3. If equals are multiplied by equals the products are 
equal. 

4. If equals are divided by equals the quotients are 
equal. 

6. Like powers and like roots of equals are equal. 

G. The whole of a quantity is greater than any of its 
parts. 

7. The whole of a quantity is equal to the sum of all 
its parts. 

8. Quantities respectively equal to the same quantity 
are equal to each other. 



a* 



10 ELEMENTARY ALGEBRA. 



SECTION II. 

ALGEBRAIC OPERATIONS. 

14. A Theorem is something to be proved. 

15. A Problem is something to be done. 

18. The Solution of a Problem in Algebra consists, — 
1st. In reducing the statement to the form of an equa- 
tion ; 

2d. In reducing the equation so as to find the value 
of the unknown quantities. 

Examples for Practice. 

1. The sum of the ages of a father and his son is 60 
years, and the age of the father is double that of the son ; 
what is the age of each ? 

It is evident that if we knew the age of the son, by- 
doubling it we should know the age of the father. Sup- 
pose we let X equal the age of the son; then 2x equals 
the age of the father; and then, by the conditions of the 
problem, a:, the son's age, plus 2^7, the father's age, equals 
60 years; or 3a: equals 60, and (Axiom 4) x, the son's 
age, is ^ of 60, or 20, and 2x, the father's age, is 40. 
Expressed algebraically, the process is as follows : — 

Let X = son's age, 
then 2x =z father's age. 
a; + 2 a; = 60, 
3 a: = 60, 

X = 20, the son's age. 
2 X = 40, the father's ago. 



DEFINITIONS. 11 

2. A horse and carriage are together worth $450 ; but 
the horse is worth twice as much as the carriage ; what 
is each worth? Ans. Carriage, S150; horse, $300. 

All problems should be verified to see if the answers 
obtained fulfil the given conditions. In each of the pre- 
ceding problems there are two conditions, or statements. 
For example, in Prob. 2 it is stated (Ist) that the horse 
and carriage are together worth $450, and (2d) that the 
horse is worth twice as much as the carriage ; both these 
statements are fulfilled by the numbers 150 and 300. 

3. The sum of two numbers is 12, and the greater is 
seven times the less ; what are the numbers ? 

4. A drover being asked how many sheep he had, said 
that if he had ten times as many more, he should have 
440 ; how many had he ? 

5. A father and son have property of the value of 
$8015, and the father's share is four times the son's; 
what is the share of each ? 

Ans. Father's, $6412; son's, $1603. 

6. A farmer has a horse, a cow, and a sheep ; the horse 
is worth twice as much as the cow, and the cow twice 
as much as the sheep, and all together are worth $490 ; 
how much is each worth ? 

OPERATION. 

Let X = the price of the sheep, 

then 2 a: = " " " " cow, 

and 4 a; = ^* '' " *' horse ; 
and their sum Tar = 490, 

a; = 70, the price of the sheep, 

and 2 a: = 140, " " " " cow, 

and 4x = 280, " '* " " horse. 



12 ELEMENTARY ALGEBRA. 

7. A man has three horses which are together worth 
$540, and their values are as the numbers 1, 2, and 3; 
what are the respective values ? 

Let a:, 2 a:, and 3x represent the respective values. 

Ans. $90, $180, and $2*70. 

8. A man has three pastures, containing 360 sheep, 
and the numbers in each are as the numbers 1, 3, and 5 ; 
how many are there in each ? 

9. Divide 63 into three parts, in the proportion of 2, 
3, and 4. 

Let 2x, 3 a;, and 4 a: represent the parts. 

10. A man sold an equal number of oxen, cows, and 
sheep for $ 1500 ; for an ox he received twice as much 
as for a cow, and for a cow eight times as much as for 
a sheep, and for each sheep $ 6 ; how many of each did 
he sell, and what did he receive for all the oxen ? 

Ans. 10 of each, and for the oxen, $ 960. 

11. Three orchards bore 812 bushels of apples ; the first 
bore three times as many as the second, and the third 
bore as many as the other two ; how many bushels did 
each bear? 

12. A boy spent $4 in oranges, pears, and apples; he 
bought twice as many pears and five times as many apples 
as oranges ; he paid 4 cents for each pear, 3 for each 
orange, and 1 for each apple ; how many of each did he 
buy, and how much did he spend for oranges ? how much 
for pears, and how much for apples ? 

. (25 oranges, 50 pears, and 125 apples. 

( Spent for oranges, $0.75 ; pears, $2 ; apples, $1.25. 

13. A farmer hired a man and two boys to do a piece 
of work ; to the man he paid $ 12, to one boy $ 6, and 
to the other $ 4 per week ; they all worked the same 
time, and received $ 264 ; how many weeks did they 
work ? Ans. 12 weeks. 



DEnXITIOXS. IS 

14. Three men, A, B, and C, agreed to build a piece 
of wall for S 99 ; A could build 7 rods, and B 6, while 
C could build 5 ; how much should each receive ? 

15. Four boys, A, B, C, and D, in counting their money, 
found they had together $1.98, and that B had twice as 
much as A, C afi much as A and B, and.D as much as 
B and C ; how much had each ? 

Ans. A 18 cents, B 36, C 54, and D 90. 

16. It is required to divide a quantity, represented by 
a, into two parts, one of which is double the other. 

OPERATION. 

Let X = one part, 
then 2x = the other part . 

Sx = a, 

X = -, one part, 

o 

2x = -—, the other part. 

o 

17. If in the preceding example a = 24, what are the 
required parts ? 

^""- 3 == T = ^' ^""^ T = T = •"• 

18. It is required to divide c into three parts so that 

the first shall be one half of the second and one fifth of 

the third. . c 2r , •'>^ 

Ans. -, — , and — . 

19. Divide n into three parts, so that the first part 
shall be one third the second and one seventh of the 
third. 

20. A is one half as old as B, and B is one third as 
old as C, and the sum of their ages is p ; what is the 
age of each ? ^^^ ^,^ P g,^ 2p_ ^^^ ^'s '^. 



14 ELEMENTARY ALGEBRA. 



SECTION III. 

DEFINITIONS AND NOTATION. 

[Continued from Section I.] 

17. The last letters of the alphabet, x, y, z, &c., are 
used in algebraic processes to represent unknown quanti- 
ties, and the first letters, a, b, c, &c., are often used to 
represent known quantities. 

Numerical Quantities are those expressed by figures, as 
4, 6, 9. 

Literal Quantities are those expressed by letters, as 
a, X, y. 

Mixed Quantities are those expressed by both figures 
and letters, as 3 a, 4 a;. 

18. The sign plus, -f-' i^ called the positive or affirm- 
ative sign, and the quantity before which it stands a 'pos- 
itive or affirmative quantity. If no sign stands before a 
quantity, -|- is always understood. 

19. The sign minus, — , is called the negative sign, and 
the quantity before which it stands, a negative quantity. 

20. Sometimes both + and — are prefixed to a quan- 
tity, and the sign and quantity are both said to be am- 
biguous; thus, 8 zi= 3 = 11 or 6, and a z^h ::= a -\-hf 
or a — 6, according to circumstances. 

21. The words plus and minus, positive and negative, 
and the signs -f- and — , have a merely relative signifi- 
cation ; thus, the navigator and the surveyor always rep- 
resent their northward and eastward progress by the sign 
+, and their southward and westward progress by the 
sign — , though, in the nature of things, there is nothing 
to prevent representing northings and eastings by — , 
and southings and westings by +• So if a man's prop- 



DEFnaiiONS. 16 

erty is considered positive, his gains should also be con- 
sidered positive, while his debts and his losses should be 
considered negative; thus, suppose that I have a farm 
worth $6000 and other property worth $3000 and that 
I owe SIOOO, then the net value of my estate is S5000 
+ $3000 — $1000 = $7000. Again, suppose my farm 
is worth $5000 and my other property $3000, while 
I owe S 12000, then my net estate is worth $5000 
_|_ S3000 — $12000 = — $4000, i. e. I am worth 

— $4000, or, in other words, I owe $4000 more than I 
can pay. From this last illustration we see that the sign 

— may be placed before a quantity standing alone, and 
it then merely signifies that the quantity is negative, 
without determining what it is to be subtracted from. 

22. The Terms of an algebraic expression are the quan- 
tities which are separated from each other by the signs 
+ or — ; thus, in the equation 4a — b =i Sx -{- c — 1 y, 
the first member consists of the two terms 4a and — 6, 
and the second of the three terms 3 x, c, and — 7 y. 

23. A CoEFFiCTEXT is a number or letter prefixed to a 
quantity to show how many times that quantity is to be 
taken; thus, in the expression ix, which equals x -\- x 
-\- X -{- X, the 4 is the coefficient of x ; so in dab, which 
equals ab -\- ab -\- ab, 3 is the coefficient of a 6; in 4 a 6, 
4 a may be considered the coefficient of b, or 4 6 the co- 
efficient of a, or a the coefficient of 45. 

Coefficients maybe numerical or literal or mixed; thus, 
in 4 a 6, 4 is the numerical coefficient of ab, a is the lit- 
eral coefficient of 4 6, 4 a is the mixed coefficient of 6. 

If no numerical coefficient is expressed, a unit is un- 
derstood ; thus, X is the same as Ix, be as 16c. 

24. An Index or Exponent is a number or letter placed 
after and a little above a quantity to show how many times 
that quantity is to be taken as a factor; thus, in the ex- 



16 ELEMENTARY ALGEBRA. 

pression b^, which equals 6 X & X &, the 8 is the index or 
exponent of the power to which b is to be raised, and it 
indicates that b is to be used as a factor 3 times. 

An exponent, like a coefficient, may be numerical, lit- 
eral, or mixed ; thus, ar^, x^, x^"^, &c. 

If no exponent is written, a' unit is understood ; thus 
b = b^, a =z a\ &c. 

Coefficients and Exponents must be careful)}'- distin- 
guished from each other. A Coefficient shows the num- 
ber of times a quantity is taken to make up a given 
sum ; an Exponent shows how many times a quantity is 
taken as a factor to make up a given product ; thus 
4:X'=x-\-X'\-x-^x, and x^=zxy,xy^xy(^x. 

25* The product obtained by taking a quantity as a 
factor a given number of times is called a power, and 
the exponent shows the number of times the quantity is 
taken. 

26. A Root of any quantity is a quantity which, taken 
as a factor a given number of times, will produce the 
given quantity. 

A Root is indicated by the radical sign, \/, or by a 
fractional exponent. When the radical sign, \/> is used, 
the index of the root is written at the top of the sign, 
though the index denoting the second or square root is 
generally omitted ; thus, 

^/ X, or x^y means the second root of x ; 
^Ic, or a:^ " '' third '' " x, &c. 

Every quantity is considered to be both the first power 
and the first root of itself. 

27t The Reciprocal of a quantity is a unit divided by 
that quantity. Thus, the reciprocal of 6 is -, and of a;, -. 



DEFINITIONS. 17 

28« A Monomial is a single term; as a, or 3a*, or 
5bxy. 

29. A Polynomial is a number of terms connected 
with each other by the signs plus or minus ; sls x -\- y, 
or da -\- 4:X — *lahy. 

30* A Binomial is a polynomial of two terms ; as 
3x + 3y, or a: — 2/. 

31* A Residual is a binomial in which the two terms 
are connected by the minus sign, as a: — y. 

32t Similar Terms are those which have the mme powers 
of tJie same letters, as x and 3 a:, or 5ax^ and — 2ax'^. 
But X and ar, or 5 a and 5 b, are dissimilar, 

33. The Degree of a term is denoted by the sum of 
the exponents of all the literal factors. Thus, 2 a is of 
the first degree ; 3 a^ and 4 fl 6 are of the second de- 
gree ; and 6 a' x* is of the seventh degree. 

34. Homogeneous Terms are those of the same degree. 
Thus, 4a^ar, 3a6c, xry, are homogeneous with each other. 

3^. To find the numerical value of an algebraic expres- 
sion when the literal quantities are known, we must sub- 
stitute the given values for the letters, and perform the 
operations indicated by the signs. 

The numerical value of 7 a — b* -\- c^ when a = 4, 
6 = 2, and c = 6 is 7 X 4 — 2* + 6^ = 28 — 16 
+ 25 = 37. 

Examples. 
Find the numerical values of the following expressions, 
when rt = 2, 6=13, c = 4, d = 15, m = 5, and n = 7. 

1. a + b — c + 2d. Ans. 41. 

2. a* + Sbc — 2cd. Ans. 40. 

3. i^^lt^* Ans. 219. 



18 ELEMENTARY ALGEBRA. 

4. m^ — 2mn + n^ 

6. ^ - (^ - ^')- 

6. (a" _ c + 6) (wi + n). 

Y. ^' ~ "•'' X {d — m + n). Ans. 86Y. 



\/c 4" V^^ + c — \/bm. Ans. 1. 



9. 3a\/& — c X 4.71 \/2bm, 

10. , ^ . Ans. 4. 

d — 2 c 



11. \^d — n + ^/1 n. 

12. (6 — fl) (cZ — c) — m. Ans. 116. 

13. 13 (4.d) + Ad — la. 



14. 4a6 + \/lOOc — ^d — n. Ans. 122. 

15. 4a s^QOl + ba'b^ 

16. 6 — a — (d — n). Ans. 3. 

17. b — a — d — n. Ans. — 11. 

18. (b — a) {d — n). Ans. 88. 

19. (b — a) d — n. Ans. 158. 



20. a + b^lO {d — m) + 14 V'c. 
36* Write in algebraic form : — 

1. The sum of a and b minus the difference of m 
and n. (m > n.) 

2. Four times the square root of the sum of a, b, and c. 

3. Six times the product of the sum and difference of 
c and d. {c >- d.) 

4. Five times the cube root of the sum of a, m, and n. 

5. The sum of m and n divided by their difference. 

6. The fourth power of the difference between a and w, 



ADDITION. 



19 



SECTION IV. 

ADDITION. 

37. Addition in Algebra is the process of finding the 
aggregate or sum of several quantities. 

For convenience, the subject is presented under three 
cases. 

CASE I. 

38. When the terms are similar and have like signs. 
1. Charles has 6 apples, James 4 apples, and William 

5 apples ; how many apples have they all ? 



6 apples, 
4 apples, 
6 apples. 



OPERATION. 

or, letting a 
represent - 
one apple, 


6 a 

4 a 

5 a 


It is evident that just as 
6 apples and 4 apples and 
6 apples added together 
make 15 apples, so 6 a and 


15 a 


cr naake 15 a. 



15 apples, 

In the same way — 6 a and — 4 a and — 5 a are 
equal together to — 15 a. 

Therefore, when the terms are similar and have like 



RULE. 

Add the coefficients, and to their sum annex the common 
letter or letters, and prefix the common sign. 



(2.) 


(3.) 


(4.) 


(5.) 


(6.) 


0') 


b ax 


3a2 


4x 


6y 


— Zj^ 


— bby 


Sax 


4a« 


X 


lOij 


— 2ar' 


— 26y 


4ax 


la' 


5x 


y 


— Ix" 


- hy 


2ax 


3a« 


3x 


2y 


— 4.x' 


- hy 



19 ax 



13 X 



-^16x» 



20 



ELEMENTARY ALGEBRA. 



8. What is the sum of arc^, Saa^, 2ax^, and 4:asP? 

Ans. 10 a x^. 

9. What is the sum of 3 5a:, 4 5a:, 6bx, and bx? 

10. What is the sum of 2a:y, 6xy, lOxi/, and Sxy? 

11. What is the sum of — Ixz, — xz, — 4a:z, and 

— xz? Ans. — 13x2;. 

12. What, is the sum of —2 5, —3 5, —6 5, and 

— 35? 

13. What is the sum of — a5c, — 3a5c, — 4a5c, 
and — a be? 

CASE II. 

39. When the terms are similar and have unlike signs. 

1. A man earns T dollars one week, and the next week 
earns nothing and spends 4 dollars, and the next week 
earns 6 dollars, and the fourth week earns nothing and 
spends 3 dollars ; how much money has he left at the 
end of the fourth week ? 

If what he earns is indicated by -[-, then what he 
spends will be indicated by — , and the example will 
appear as follows : — 

OPERATION. Earning 7 dollars and 

then spending 4 dollars, 
the man would have 3 
dollars left^ then earn- 
ing 6 dollars, he would 
have 9 dollars; then 
spending 3 dollars, he 
would have left 6 dol- 
lars ; or he earns in all 7 dollars -j- 6 dollars =13 dollars ; and spends 
4 dollars -|- 3 dollars = 7 dollars; and therefore has left the differ- 
ence between 13 dollars and 7 dollars == 6 dollars; hence the sum 
of -f- 7 d, — 4 d, -{- G d, and — 3 d is -\- 6 d. 

Therefore, when the terms are similar, and have unlike 
signs : 



+ 7 dollars, 




-\-*ld 


• — 4 dollars. 


or, letting d 


— 4:d 


+ 6 dollars, 


. represent ^ 


+ 6d 


— 3 dollars, 


one dollar, 


— 3d 


+ 6 dollars. 


+ ed 



ADDITION. 21 

RULE. 
Find the difference between the sum of the coefficients of 
the positive terms, and the sum of the coefficients of Uie neg- 
ative termSf and to this difference annex the common letter 
or letters, and prefix the sign of the greater sum. 



(2.) 


(3.) 


(4) 


(5.) 


3xy 


^y" 


13o6c2 


U^y 


xy 


-2i/» 


6a6c» — 


13x2 1/ 


— 5a:t/ 


ly" 


— ahc" 


lOx^y 


Ixy 


-31/^ 


— Sabc' — 


^r^y 


— 2x1/ 


Uy" 


4a6c2 


24x^3/ 


4x1/ 


Uabc" 






(6.) 


(T.) 






25x1/2 


8(x + i/) 




— 


- 60 X 1/ z 


-4(x + i/) 






10x1/2 


n^ + y) 




— 


-6txi/2 


-3(x + i/) 






8x1/2 


- (^ + 2/) 





— 74x1/2 T (x + y) 

8. Find the sum of 8 xV, — Hx^y", ITx^y*, and — x^^. 

9. Find the sum of T(x+y), 8(x + y), — (x + y), 
and 4 (x + y). Ans. 18 (x + y). 

10. Find the sum of — a x^, + a x*, — 10 a x^ + 25 a x^, 
and — 13ax2. 

11. Find the sum of 21 a b, — 34 a i, — 150 ab, 21 a b, 
and — 13 a i. Ans. — 143 a b. 

12. Find the sum of ax*, — 14 a x", IT ax*, — ax*, 
44 a X*, and — a x*. 

13. Find the sum of 17 (a + b), — (a + b), (a + b), 
and — 13 (a + 6). Ans. 4 (a + 6). 



22 ELEMENTARY ALGEBRA. 

CASE III. 
40. To find the sum of any algebraic quantities. 

The sum of 5 a and 6 & is neither 11a, nor \lh, and 
can only be expressed in the form of b a -\- Qb, or 6 5 
-j- 5 a ; and the sum of 5 « and — 45 is 5a — 45; but in 
finding the sum of 5 a, 6 5, 5 a, and — 4 5, the a's can be 
added together by Case I., and the 5's by Case II., and 
the two results connected by the proper sign ; thus, 5 a 
4-65 + 5a — 45r=10a + 2 5. 

1. Find the sum of 6 c?, —25, x, Bi/, 5 x, —3 5, 35c 
+ 4 c?, 5x, T 5 + 2 X, and — 3 5 c. 

OPERATION. Yor convenience, simi- 

^d — 25-1- X -\- Sy -{- Sbc lar terms are written un- 

4.d—Sb^6x —3 be ^^^ ^^ch other ; then by 

+ Y5-|-5a: 

-\-2x 



Case I. the first column 
at the left is added; the 
second by Case II., and 
10c?-l-2 5-f 13a; + 3^^ so on ;+ 35c and — 35c 

cancel. 

This case includes the two preceding cases, and hence 
to find the sum of any algebraic quantities : 

RULE. 

Write similar terms under each other, find the sum of 
each column, and connect the several sums with their proper 
signs. 

(2.) 
4a;— Ya-t-3y — 45-t-3z 
6a— ij -\- 4:b — 2z 
4:a — 2i/-\-Sb— z 
— 3a — 85 — 10 c 

4x —10c 



ADDITION. 23 

(3.) 

— 36+ 3c — n \/x + y 
— 10c-|-8\/^ — y 

7 a 4- 6— 10c + 6\/x 

4. Add together *l s/~x, — 8.r, 7 x'*, — 6\/^ 4a:*. 

— 8 a:, 4 a:, and 7 a:'*, — ^ \^ x. 

Ads. 18 ar*— \2x — *l\/x. 

5. Add together Zax — 4a6 + 2a7y, *l ah -\- bx — 4a, 
*l xy — 3 a + 4 X, and -{- ahc — ax -\- 6a: y. 

6. Add together 7a: — 3ay — 5a6-|-4c, 3aa: + 4ar 
-}- 5 a i — 5 c, and 3c — 3ax-f- *?y4"<^- 

Ans. llx — 3ay + 3c-|-7y. 

7. Add together 5 a — 32 + 7x + 4aa: — 3a^ bah 

— 5a + 22 — 4aa: + 4, and 6 — 2 a6 + 3a: + 4y -f 
4 ax. 

8. Add together ^xy -\- Qxz — 6mn-4-4n, 4mn — 
3xy-|-2n — 8m n, — 6a:z + 47i — 3a:y4"6, and 10 win 

— lOn + 3 — 9. Aim. 0. 

9. Add together 8am+19na:— 55 3 + c, — 19u + 
14 6 — 16c+y, and 18 n a: — 44 am + 15 u — 4y. 

10. Add together 17 aa:« + 19 aar^ — 14 aa:* + 16 aa:^, 
13 ax' —5 ax* + 6 aar» — 10 ax\ and Uax* + 17 a x^ 

— 3ax'^+15ax^ Ans. 71 ax^ + lOax"** — 5ax\ 

11. Add together m -^ n — 4a + 6c — 7y, 8c — 4m 
+ 3n — 5a + 3c, 7a— 17c + 7y— 10m— 6n, and 
14n — 8a — 7c+ lOy — 8 m. 

12. Add together 8axy+17 6xy — 16cxy — 9axy, 
16 6xy — 18 cxy + lOaxy — 14 ax 2, 16cxy + 25axy 

— 7 6xy + 25cxy, and lOaxar + Zhxy — lOcxy + 
4ax2. Ans. 34axy + 296xy — Sexy. 



24 ELEMENTARY ALGEBRA. 

13. Add together S{x -\- i/), — 4:{x -\- y), and 1 (x -{- y). 

Ans. 6 (x -f- y). 
14 Add together 5 (2 x + y — 3 z), and —2(2x-\-y 

— 3z). 

Note. — If several terms have a common letter or letters, the 
sum of their coefficients may be placed in parenthesis, and the com' 
mon letter or letters annexed; thus, 

6a:-f8a; — 5ic=(6 + 8 — 5)a:; 
ax -^ dh X — 2 c X = (a -\- ^h — 2c)x; 
hcxy-\-adxy — ac xy = {h c -\-ad — a c) xy. 

16. Add together ax — bx -\- Zx, and 2ax -\- 4:bx — x. 

Ans. (3a + 36 + 2) x. 

16. Add together by — Scy -\-6ay, and cy -\-4:by 

— 2 ay. 

11. Add together 2xy — axy, and 6xy — Saxy. 

Ans. (8 — 4 a) xy. 

18. Add together 1{Bx-{-5y)-\-Sa — 6x + Sab, 3x 
+ 5 (3a: +5y) -{-la —bab, and 8x +2(Sx -\- 5y) 
^la — Sab. Ans. 4Yx+ TOy + 3a. 

41. From what has gone before, it will be seen that 
addition in Algebra differs from addition in Arithmetic. 
In Arithmetic the quantities to be added are always con- 
sidered positive ; while in Algebra both positive and neg- 
ative quantities are introduced. In Arithmetic addition 
always implies augmentation ; while in Algebra the sum 
may be numerically less than any of the. quantities added ; 
thus, the sum of 10 a: and — 8 a; is 2x, which is the 
numerical difference of the two quantities. 



SUBTRACTION. 25 

SECTION V. 

SUBTRACTION. 

42. Subtraction in Algebra is finding the difference 
between two quantities. 

1. John has 6 apples and James has 2 apples ; how 
many more has John than James ? 
Let a represent one apple, and we have 

6a 
2a 

4a 



., or 6a — 2a = 4a. 



2. During a certain day A made 9 dollars and B lost 
6 dollars ; what was the difference in the profits of A 
and B for the day ? If gain is considered -|-> ^^^^ loss 
must be considered — , and letting d represent one dol- 
lar, it is required to take — 6 c? from 9 d. 



OPERATION. 

9rf 
— 6d 



It is evident that the difference be- 
tween A's and B's profits for the day 

is 9 c? + 6 c/ = 15rf; that is, 9 d — 

I5d (-6<0 = 9rf+6rf=15rf. 

Hence it appears that, as addition does not always im- 
ply augmentation, so subtraction does not always imply 
diminution. 

Subtracting a positive quantity is equivalent to adding an 
equal negative quantify; and subtracting a negative quan- 
tity is equivalent to adding an equal positive quantity. 

Suppose I am worth $1000; it matters not whether a 
thief steals $400 from me, or a rogue having the author- 
ity involves me in debt $400 for a worthless article ; for 
2 



26 ELEMENTARY ALGEBRA. 

in either case I shall be worth only $600. The thief sub- 
tracts a positive quantity ; the rogue adds a negative quan- 
tity. 

Again, suppose I have $1000 in my possession, but 
owe $400 : it is immaterial to me whether a friend pays 
the debt of $400 or gives me $400 ; for in either case I 
shall be worth $1000. In the former case the friend 
subtracts a negative quantity ; in the latter, he adds a pos- 
itive. Or, to make the proof general : 

1st. Suppose + 5 to be taken from a -\- h 

the result will be a ; 

and adding — h to a -\- h v7Q have a -\- h — h, 

which is, as before, equal to a. 

2d. Suppose — h to be taken from a — b 

the result will be a ; 

and adding -\- b to a — b we have a — b -\- b, 

which is, as before, equal to a. 

3. Subtract b -\- c from a. 

OPERATION. I subtracted from a gives 

a — (b -X- c) :=. a — b — c a — b; but in subtracting b 

we have subtracted too small 
a quantity by c, and therefore the remainder is too great by f, and 
the remainder sought is a — b — c. 

4. Subtract b — c from a. 

OPERATION. In subtracting b from a we 

a — (b — c)=a — 5-j-c subtract a quantity too great 

by c ; therefore the remainder 
(a — b) would be just so much too small, and the remainder sought 
is a — 6 -|- c. 

43* By examining the examples just given it will be 
seen that in every case the sign of each term of the. 
subtrahend is changed, and that the subsequent process 
is precisely the same as in addition ; hence, for subtrac- 
tion in Algebra we have the following 



SUBTRACTION. 27 

RULE. 
Change (he sign of each term of (he subtrahend from -f- 
io — , or — to -f-> or suppose each to he changed, and then 
proceed as in addition. 





(!•) 


(2.) 


(3.) 


(4.) 


(5.) 


(6.) 


0-) 


Min. 


9 


9 


9 


9 


9 


9 


9 


Sub. 


9 


6 


3 





— 3 


— 6 


— 9 



Rem. 3 6 9 12 15 18 

In examples 1-7, the minuend remaining the same while 
the subtrahend becomes in each 3 less, the remainder in 
each is 3 greater than in the preceding. 

(8.) (9.) (10.) (11.) (12.) (13.) (14.) 
Min. 9 6 3 0—3—6—9 

Sub. 9 9 9 9 9 9 9 



Rem. —3 —6 —9 —12 —15 —18 

In examples 8-14, the minuend in each becoming 3 less 
■while the subtrahend remains the same, the remainder in 
each is 3 less than in the preceding. 

(15.) (16.) (17.) (18.) (19.) (20.) (21.) 
Min. 9 6 3 —3 —6 —9 

Sub. 9 6 3 —^ — ^ —9 

Rem. 

In examples 15-21, both minuend and subtrahend de- 
creasing by 3, the remainder remains the same. 

(1.) (2.) (3.) (4.) (5.) (6.) 

Min. 26x 27axy — 13a6 —18c 49xy —4386 

Sub. 10a: — 4axy 4a6 — 6c — 25xy 27 6 

Rem. 16x Zlaxy —llab — 12tf 



28 ELEMENTARY ALGEBRA. 

(1.) (8.) (9.) (10.) (11.) (12.) 

Min. lOx — 4:axy 4,ah — 6c — 2bxy . 21h 

Sub. 26x 21axi/ —ISab — > 18 c 4:9x1/ —438^ 

Rem. — 16a: — Slaxi/ 11 ab 12c 

(13.) (14.) 

Min. 6 a:— 14^ + 3 2r 1 a+18b — 10c 

Sub. 3a:+3y+;2 -_25 5+6c — 8</ 

Rem. Sx—l1t/-j-2z 1a-\-4:3b — 16c + Sd 

15. From 28 a: take — 11 x. Ans. 45 a:. 

16. From — 34Y a take 223 a. Ans. — 570 a. 
11. From — T6y take — 33 y. Ans. — 43 y. 

18. From Ub take —150 5. Ans. 194 5. 

19. From — 411 c take 984 c. 

20. From — 84;^ take — 117 z. 

21. From 17 a a: — 18 5c + 44a:y take 25 6c — 14 a: 3^ 
4- 20 a. Ans. 17 a a: — 43 J c + 58 a:y — 20 a. 

22. From 384 a:— 74y-f- 18 c take 118 a: + 743^ — 27 c. 

Ans. 266 a:— 148 3/ + 45 c. 

23. From x^ — f + x^ — 10 a^ take x'^ + 4.f — x^-^ 
4.7?. Ans. 2a:* — 6a:^ — 5y. 

24. From Qaby — 4a:y-j-3a:2; take — 4,aby — 3a:;3 
<— 4 a: y.-^ 

25. From a:^ -f 2 a: ^^ + ^2 ^^^ ^"^ _ 2xy -{- y"^. 

26. From x^ -\- 2 x y -{- y"^ take —x^-\-2xy — y\ 

44. The subtraction of a polynomial may be indicated 
by enclosing the polynomial in a parenthesis and prefix- 
ing the sign — . 

Thus, a? -\- i^ — z^ taken from a:' — s? may be written 



SUBTRACTION. 29 

When a parenthesis with the sign minus be/ore it is re- 
moved, the sign of each term within the parenthesis must be 
changed according to the Bute for subtraction. 

Thus, ar^ — 23_(^_j_y»_2«) = 2r'' — z3_^_^_j. 

And conversely, 

A polynomial, or any number of the terms of a polyno- 
mial, can be enclosed in a parenthesis and the minus sign 
placed before the parenthesis without changing the value of 
the expression, prodding the signs of all the terms are 
changed from plus to minus or from minus to plus. 

Thus, a' ^ Ir + c" + d — X = a^ — {ty" — c"" — d -j- x). 

Note. — When the sign of the first term in the parenthesis b 
plus, the sign need not be written. (Art. 18.) 

According to this principle a polynomial can be writ- 
ten in a variety of ways. 

Thtis, a:*— 3 a:2y4- 3 a:y2— y' = ar' — (3 x2y — 3 a:y2 -{_ y3) 

= ar^ + 3x7f-(3x'f/-\-^') 

Remove the parenthesis, and reduce each of the follow- 
ing examples to its simplest form.* 

1. a^ — (2ab-{- c^). Ans. a^ — 2 a ^ — c«. 

2. x^ — 6ax -\-r^ — 6x^y—{x'-\-6ax + 2^ — ex^y). 

Ans. — 12 ax, 

3. m^ — n^-\-2x—(4:m^-\-'Sn^ — 4:c). 

4. 16a:y+ Uc — lSy— (— Uc + 27y — 16a:y). 

Ans. 32xy + 28c — 46 y. 
6. 4:3^1/— (Sxf—lx'f + Sa^i/), 
6. — (-a:«+7-25xy + /). 



80 ELEMENTAEY ALGEBRA. 

Place in parenthesis, with the sign — prefixed, without 
changing" the value of the expression, 

1. The last three terms of 1 x'^ — 14 xy — Ssr + ^y. 

Ans. 1 x''—{Uxi/-\-Sz — 4.y). 

2. The last three terms of x^ ■}- y^ — Zxy -\- ^c. 

Ans. x'^ — i^xy — y^ — 4 c). 

3. The last four terms oi4.a — 1h — Qc — ^d-\-x'^. 

4. The last four terms of a^ -\. h^ -\- c"" — d"" + a\ 

5. Write in as many forms as possible by enclosing 
two or more of the terms in parenthesis, a^ — b^ -f~ ^^ 
^d\ 

45. In subtraction, when two quantities have a com- 
mon factor their difference is the difference of the coef- 
ficients of the common factor multiplied by this factor. 

Thus, ax — hxr=z {a — V) x. 

1. From ax^ take cx^ — dx^. Ans. (a — c -\- d) x^. 

2. From 4 \/ a; take a\/ x -{- h^/ x. 

Ans. (4 — a — h) \/ x. 

3. From ax^ take hx^ — hx^. Ans. {a — h) x^ -\- h x^. 

4. From 4 a:^ — Qx take ax"^ -\- hx. 

Ans. (4 — a) x^ — (6 -|- b) x. 

6. From 6 a^ -|- 4 a^ — a take a^ x — a^y -\- az. 

Ans. (6 — x)a'-{- (4 +y) a" — {I + z) a. 

6. From ah — he take ^h -\- ex. 

1. From a"^ — bx -\- c a^ x take hx^ -\- c x — d^J x. 

8. From xy'^-^x' — x^y'' take y"" + x'^y — x'^y^. 



MULTIPLICATION. 81 

SECTION VI. 

MULTIPLICATION. 

46* Multiplication is a short method of finding the 
sum of the repetitions of a quantity. 

47. The multiplier must always be an abstract num- 
ber, and the product is always of the same nature as the 
multiplicand. 

The cost of 4 pounds of sugar at 17 cents a pound is 
n cents taken, not 4 pounds times, but 4 times ; and 
the product is of the same denomination as the multi- 
plicand 17, viz. cents. 

In Algebra the sign of the multiplier shows whether 
the repetitions are to be added or subtracted. 

1. (+a)X(+i) = + ia; 
i. e. -\- a added 4 times is -\-a-{-a-{-a-\-a = -\-4ia. 

2. (+a)X(— 4) = -4a; 
i. e.-\- a subtracted 4 times is — a — a — a — a = — 4 a. 

3. (-a)X (+4) = -4a; 
i. e. — a added 4 times is — a — a — a — a = — 4 a. 

4. (--a)X(— 4) = + 4a; 
i.e. — a subtracted 4 times i3-{-a-\-a-\-a-{-a = -\-4:n. 

In the first and second examples the nature of the 
product is -j- ; in the first, the -[" sign of 4 shows that 
the product is to be added, and + 4:a added is + 4a; 
in the second, the — sign of 4 shows that the product 
is to be subtracted, and + 4a subtracted is — 4a. In 
the third and fourth examples the nature of the product 
ij — ; in the third, the + sign of 4 shows that the prod- 
uct is to be added, and — 4 a added is — 4 a ; in the 



82 ELEMENTARY ALGEBRA. 

fourth, the — ■ sign of 4 shows that the product is to be 
subtracted, and — 4 a subtracted is -\- 4:a. 

48. Hence in multiplication we have for the sign of 
the product the following 

RULE. 
Like signs give -\- ; unlike, — . 

Hence the products of an even number of negative fac- 
tors is positive, of an odd number, negative. 

49. Multiplication in Algebra can be presented best 
under three cases. 

CASE I. 

50. When both factors are monomials. 
. 1. Multiply 3 a by 2 5. 

OPERATION. 

3aX25 = 3X«X2x^ — 3X2X«X^ = 6aJ. 

As the product is the same in whatever order the factors are 
arranged, we have simply changed their order and united in one 
product the numerical coefficients. 

Hence, when both factors are monomials, 

RULE. 
Annex the product of the literal factors to the product 
of their coefficients, remembering that like signs give -|- 
and unlike, — . 

2. Multiply a^ by a\ 

OPERATION. 

a^X«^=(«X«X«)X(«X«) = «XaX«X«X« = a^ 
As the exponent of a quantity shows how many times it is taken 
as a factor, a' = a X « X « I and a^ = a y, a', and a^ X «^ = ot X 
a X a X « X «» and this is equal to a^ (Art. 24.) Hence, 

Powers of the same quantity are multiplied together by 
adding their exponents. 



(3.) 
Zah 


(4.) 

5x^y\ 

Zxtf 

15 rV 


MULTIPLICATION. 

(5.) (6.) 
7 a 6 _- 14 wn^ 
— %an 6an* 


33 

a.) 

— 40=6 


I2abxy 


— 66 a^ i^ 


— 84 a m n' 


4a<^»^ 



8. Multiply x^ by ar*. 

9. Multiply x^ by a/. 

10. Multiply a^ by — a. 

11. Multiply —a* by ct". 

12. Multiply — c« by — c^ 

13. Multiply 8 a:/ by Taar. 

14. Multiply 504 a^ fr'' by — 8 a« h. 

15. Multiply — 25a:y2 by AiXyz, 

16. Multiply — 411 a 6 c2 by — 3 a 5» c. 

17. Multiply together 444 xy, 3^^/, and — 2 2;. 

Ans. — 2664ar5y*2r. 

18. Multiply 4 a2 ^2 ^ c? by — 4 a J c^ rf2. 

19. Multiply — 5 a:^ by — 6 a:*. Ans. 30 ar^+". 

20. Multiply together 14 a 6 c-, — 5 d^ h c, and — 4 a J'^. 

21. Multiply 25 \/ay by 3 6a:. Ans. 1bhx\/ay. 

22. Multiply 4 (a: + y) by 3 (x + y). 

Ans. 12(x + y)-. 

Note. — Any number of tenng enclosed in a parenthesis may 
be treated as a monomial. 

23. Multiply — 12 (a^ — ^) by — 4 {a" — P). 

Ans. 48 (a« — H^K 

24. Multiply (a — x)* by (a — x)^. 

25. Multiply 4 (a + 6)"* by 2 (a + b)\ 

Ans. 8 (a + 6)""+". 

26. Multiply a» (x + z)» by a 6« (x + «). 

2* C 



34 



ELEMENTARY ALGEBRA. 



CASE II. 
51* When only one factor is a monomial 

1. Multiply 8 + 5 by 3. 

OPERATION. 

8+ 5 



8 + 


5] 


8 + 


5 


8 + 


5 


24 + 


15. 



I or 



13 
3 



24 + 15 = 39 



2. Multiply 8 — 5 by 3. 



In this example, not the 
sum of 3 repetitions of 8 
only, but of 8 and 5, is re- 
quired ; the sum of 3 repeti- 
tions of 8 = 24 ; of 3 repeti- 
tions of 5 = 15. Hence, the 
sum of 3 repetitions of 8 -[- 
5 = 24 -f- 15. 



OPERATION. 



8 — 


6] 


8 — 


5 


8 — 


5 


24 — 


15 



- or 



— 5=3 
3 3 



24 



15 =9 



The sum of 3 repetitions 
of 8 = 24 ; but it is not the 
sum of 3 repetitions of 8 that 
is required, but of a num- 
ber 5 units less than 8 ; 24, 
therefore, will have in it the 
sum of 5 units repeated 3 
times, or 15, too much; the product required, therefore, is 24 — 15. 

Therefore, 

The product of the sum is equal to the sum of the prod- 
ucts, and the product of the difference to the difference of 
the products. 

3. Multiply X -\- 7/ — z by a, 

OPERATION. 

X U- y z "^^^ ^^^ ^^ *^® repetitions 

of a; a times, of y a times, and 

— . . - of — z a times is a x -\- ay 

- az. 



ax 



ay 



az 



RULE. 
Multiply each term of the multiplicand by the multiplier, 
and connect the several results by their proper signs. 



MULTIPLICATION. 35 

(4.) . (5.) 

3ax — 4a: 



12 ao:' — 24 aar' + 42 aary — 4 a^x + 8 a 6a; — 4 ^r'a:. 

6. Multiply 5 rn n + 4 m'' — 6 n^ by 4 a 6. 

7. Multiply IGa^ar — 8arz + 4y by —Zxy. 

8. Multiply 6a^ — c x= + rfa: by — ar*. 

9. Multiply — 63 a:y — 14 a: — 6 ^ by — 4 z. 

Ans. 252 ary z + 56 X z + 24 r». 

10. Multiply 14 a* — 13 a'' + 12 a^ — 11 a by 4 a**. 

11. Multiply a: — 2 a + 14 by a x. 

12. Multiply 17aa: — l-kby -\- llcz by — 4:abcxyz. 

13. Multiply 21 aH^ — 3 x/ _ 4 5 c by — 9 a a;y. 

CASE III. 
52« When both factors are polynomials. 

1. Multiply 7 + 4 by 5 — 3. 

OPERATION. Multiplying 7 -f 4 by 5 is 

»y I 4 =11 taking the multiplicand 3 too 

R q o many times ; therefore, the 

true product will be found by 



35 + 20 — 21 — 12 =22 subtracting 3 (7 -f 4) from 

5(7+4). 

2. Multiply X — y by a -\- b. 

OPERATION. a times x-^y = ax — ay\ but 

X y X — y is to be taken, not a times 

^ I J only, but a-\-b times; therefore, 

a (a: — y) is too small by ft (a: — y)\ 

ax — a y + 6 X — by and the product required is a x — 

ay -\-hx — by. Hence, 

RULE. 

Multiply each term of the multiplicand by each term of 
the multiplier, and find the sum of the several products. 



86 ELEMESTAEY ALGEBEA. 

(3.) . (4.) 

a^ -\- 2 a X -\- x^ x^ -\- a b 

a — X xy — ah 



c^ -\-2'C?x-\- ax^ 7?y^ '\- ahxy . 

— a^ X — 2ax^ — x^ — abxy — a^y^ 

a^ -\- a^ x — ax^ — x^ x^y"^ — a^ IP- 

5. Multiply a2 _|_ j2 _ ^2 ^^ ^2 _j_ ^2, 

Alls, a" + a^ 52 + 52 c" — c\ 

6. Multiply x"^ — 2xy -\- y"^ by x"^ -\- y'^. 

Ads. x^ — 2x^y^2 x'y'' — 2xy^ + y\ 

1, Multiply 4 a^ — 2 an + 3 a2 52 by 2a'' — 2b\ 

Ans. 8 a^ — 4 a^ — 2 a* ^>2 _|. 4 ^3 J3 _ g ^2 54^ 

8. Multiply x^ + 2x^ + Sx''-]-2x+lhjx'' — 2x+l. 

9. Multiply x^-{-y^-\-z^ — xy — xz — yz hy x -\-y -\-z, 

Ans. x^ + 5/^ + 2'^ — 3 X y z. 

10. Multiply 4 a;5 — T ic^ + 10 ar' — 13 a;2 by 3 a: — 2. 

11. Multiply a^ + a2 — 1 by a" — 1. 

12. Multiply x^+lax—Ua^ by a.- —la. 

Ans. ar^ — 49a2a:— 14a3a; + 98a*. 

13. Multiply X -\- y — a by x — y -{- a. 

14. Multiply a" + 5'* by a"* — b"". 

Ans. a "'+" + a*" 6" — a" 6"* — 5'" + ^ 

15. Multiply Ixy— 14. x" y"^ -\- 21 x"" f by Gary — 3. 

Ans. — 21 xy + Ux"" f — 14.1 a? f + l2Qx^y\ 

16. Multiply 6 an — 9 a S2 __ 12 aH^ by 2 a & — 3 5^. 
Ans. 12 a^ }? — 36 a^ 6^ _ 24 a^ Z.» + 2Y a i* + 36 a" b\ 

V\. Multiply x^ — x^ -\- x^ — x -\- \ by a; + 1. 

Ans. x^ + 1. 

18. Multiply x^ — x^-\-x} — x-\-\ by a: — 1. 

Ans. x^ — 2x''^2x^ — 2i?-\-2x—\. 

19. Multiply X* + ar^ + a;2 + a: + 1 by ar + 1. 

20. Multiply a;* + a^ + a:^ + a; + 1 by cc — 1. 



DIVISION. 87 



SECTION VII. 

DIVISION. 

53. Division is finding a quotient which, multiplied by 
the divisor, will produce the dividend. 

In accordance with this definition and the Rule in 
Art. 48, the sign of the quotient must be -|- when the 
divisor and the dividend have like signs ; — when the 
divisor and the dividend have unlike signs ; i. e. in di- 
vision as in multiplication we have for the signs the fol- 
lowing 

RULE. 
Like signs give -f- > unlike, — . 

CASE I. 
54* When the divisor and dividend are both monomials. 

1. Divide 6ab by 2b. 

orERATiON. Thg coefficient of the quotient must 

6ab-7-2b = Sa be a number which, multiplied by 2, 

the coefficient of the divisor, will give 

6, the coefficient of the dividend ; i. e. 3 : and the literal part of 

the quotient must be a quantity which, multiplied by &, will give ab\ 

i. e. a: the quotient required, therefore, is 3 a. 

Hence, for division of monomials, 

RULE. 

Annex the quotient of the literal quantities to the quotient 
of their coefficients, remembering that like signs give -\- and 
unlike, — . 



38 ELEMENTARY ALGEBRA. 

2. Divide a^ by a\ 

OPERATION. 

a« -T- a^ = a^ For a' X a' =^ a^ (Art 50.) Hence, 

Poivers of the same quantity are divided by each other by 
subtracting the exponent of the divisor from that of the 
dividend. 



(3.) 

21x^y'' 

9 xy 

(5.) 
•^2lQa?y 

4.Qx''y 



3xy 



= — 6a; 



(4.) 






4:8a^xy 
— 16aa; 


^-S 


ay 


(6.) 






— 16a2 


x^yz__ 




— 4 axyz 






* 




f- 


Ans. — 


■Sxy. 


2iahx. 


Ans. 


6ab. 



1. Divide S4:aby^ by 2 ay. 

8. Divide 297 icV by — 99a:y2. 

9. Divide — 14,xy^z by 2a:2/. 

10. Divide —UiaH^x by — 24«5a:. 

11. Divide ax* by czar'. 

12. Divide 8 a:« by — 8 x\ 

13. Divide — 210a;''y by ^2 x^ y. 

14. Divide —210abx by — 135 a i a;. 

15. Divide —4t14:aH^c^ by 158a53c. 

16. Divide x"" by ar". Ans. a:"""". 
11. Divide 14 a"* a:" by — 1 a"" x. Ans. — 2 a"* ""a;" 

18. Divide — UT a^ b* c^ d' by SSaHc d\ 

19. Divide 12 (x -\- yf by 4.{x-\-y). Ans. 3 (a: + 2^). 

20. Divide —21 {a — by by 9(a — 5)^. 

21. Divide {b — cy by (b — cy. Ans. (5 — cf. 

22. Divide 14 (x — yf by t (a: — yy. 



— 1 



DIVISION. 39 

CASE II. 
55» When the divisor only is a monomial. 

I. Divide ax -{- at/ -{- az by a. 

OPBRATiox. In the multiplication of a poly- 

a)ax-^ay-\-az nomial by a mononiial, each term 

jp I -. I ^ of the multiplicand is multiplied 

by the multiplier ; and theiv fore 

we divide each term of the dividend ax -\- ay -\' az by the divisor 

a, and connect the partial quotients by their proper signs. Uence, 

RULE. 
Divide each temi of the dividend by the divisor, and con- 
nect the several results by their proper signs. 

(2.) (3.) 

3 a ) 6 a x^ — 24: aa? — 5 a:« y ) — 15 ai^y — 25 ar^y 

2 2r»— 8ar» 3x + 5 

4. Divide 12 a ar* — 24 a a:^ + 42 a a:y by 3 ax, 
6. Divide — 4:a^x-\-Sabx — 4 6^ar by — 4a:. 
6. Divide 6 a^x* — 12 a^x^ + 15 a*ar« by 3 a^x\ 

Ans. 2 a;* — 4 a x -[- 5 a* a:*. 
1. Divide 12a*y« — 16 a^y* + 20 a«y< — 28 aV by 
4aV. 

8. Divide — 6 ar» + 10 x^ — 15 x by — 5 x. 

9. Divide 273 (a + xy — 91 (a + x) by 91 (a + x). 

Ans. 3(a + x) — l = 3a + 3x— 1. 
10. Divide 20a6c — 4ac -f 8acd— 12 a^ c^ by -^ 4: a c. 

II. Divide 16 a^x*— 32«2x^ + 48 «*x* by 16 a* x*. 

12. DivideT2x«y« — 36x«y»~64x»y«2«by — 18x2y^ 

Ans. --4 + 2x?/+3x2«. 

13. Divide 18ax»y— 54x^2/*+ 108cx^y« by 9x*y. 

14. Divide 40 a*b^ + 8 a' 6^ — 96 a» i«x« by 8 a* b\ 

15. Divide 39x*2« — 65aa:«2* + ISx'z* by — ISx^z^. 



40 ELEMENTARY ALGEBRA. 

CASE III. 
56. When the divisor and dividend are both polyno- 
mials. 

1. Divide :t? — Zx'y -\-Zxy' -^ f by x^ — 'lxy-^- y^. 

OPERATION. 

7? — 2 a; ^ -f- ^^) ic^ — Zx^ y -\- ^xy^ — ^ (a^ — y 
a^ — 2 x'^ y -\- X y"^ 

— x"^ y -{- 2 X y^ — if 
-- x''y + 2xy^^f 

The divisor and dividend are arranged in the order of the powers 
of X, beginning with the highest power, a;^, the highest power of x 

in the dividend, must be the product of the highest power of x in 

x^ 
the quotient and x^ in the divisor ; therefore, —^ = x must be the 

highest power of x in the quotient. The divisor x^ — 2xy -\- y^ mul- 
tiplied by X must give several of the partial products which would 
be produced were the divisor multiplied by the whole quotient. 
When (x' — 2xy -\- if) X = x^ — ^x"^ y -\- xif is subtracted from 
the dividend, the remainder must be the product of the divisor and 
the remaining terms of the quotient ; therefore we treat the remain- 
der as a new dividend, and so continue until the dividend is ex- 
hausted. 

Hence, for the division of polynomials we have the fol- 
lowing 

RULE. 

Arrange the divisor and dividend in the order of the 
powers of one of the letters. 

Divide the first term of the dividend by the first term of 
the divisor ; the result will be the first term of the quotient. 

Multiply the whole divisor by this quotient, and subtract 
the product from the dividend. 

Consider the remainder as a new dividend, and proceed 
as before until the dividend is exhausted. 



DIVISION. 41 

Note. — If the dividend is not exactly divisible by the divisor, the 
remainder must be placed over the divisor in the form of a fraction 
and connected with the quotient by the proper sign. 

2. Divide a:* + 4y* by a:« — 2ary + 2y'. 



x*—2x'y 


+ 2^1/' 






2x>y 




4y 
ixf 






2x»y' — 
2r'y» — 


ixy' + 
4xy» + 


4y 

4y 



Note. — By multiplying the quotient and divisor together all the 
terms which appear in the process of dividing will be found in the 
partial products. 

3. Divide a:* — 1 by a: — 1. 

X — 1) X* — 1 (x^' + x' + x+l 



x» 


— 


1 










• 3^ 


— 


7^ 














x" 


— 


1 








3? 


— 


X 












X 


— 


1 










X 




1 



4. Divide ax — ay -{-hx — by -\- z by x — y, 

ax — ay 
ax — ay 



x^y)ax-^ay-\-hx^hy-\-z{a + h-\- j-^ 



hx — hy 
hx — hy 



5. Divide 2by^2}ry — Zlryz-\-Qb^y-\-byz—yz 
by 2 6 — 2. Ans. 3 h^y — by-{-y. 



42 ELEMENTARY ALGEBRA. 

6. Divide c^ -\- x^ by c -\- x. Ans. c^ — ex -\- x^. 

1. Divide a^-{-a^-{-a^x-\-ax-\-Sac-\-Schya-\-l. 

Ans. a^ -\- a X -{- 3 c. 

8. Divide a -\- b — d — ax — bx-\-dx by a-{- h — d. 

9. Divide 2 a* — 13 a^ 2/ + 11 a^y^ _ g « ?/ -f 2 ?/ by 
2 or — ciy -\- y'^. Ans. a^ — 6 a y + 2 ?/l 

10. Divide a^ — Z an -\- Z ah'' — b^ hy a^ — 2 ah -\- h\ 

11. Divide 2 a;3— 19x2+ 26a:— 19 by a: — 8. 

Ans. 2a;2 — 3a;4-2 ?-r 

' X — 8. 

12. Divide ar^ + 1 by a: + 1. 

Ans. X* — x^ -\- x' — a: -|- 1. 

13. Divide a;« — 1 by a; — 1. 

Ans. a:^ 4" ^* ~1~ ^^ + ^^ "t" ^ ~l~ 1- 

14. Divide x^ — 1 by a: -j- 1. 

15. Divide -x^ — ^ by x — y. 

Ans. x^ -\- x^ y -\- x^ y'^ -\- X ^ -\- y^. 

16. Divide a^ — x^ by a — x. 

17. Divide m^ — n^ hy m' -\- m n -\- n^. 

Ans. w^ — m^n -\- mn^ — n*, 

18. Divide 4 a* — 9 a^ + 6 a — 3 by 2 a^ + 3 a — 1. 

19. Divide a:* + 4x2^2 _|_ 3^4 |3y ^_^ 2 y. 

20. Divide a^ — a^ x"" -\- 2 a a? — x"" by a^ — ax-{- x\ 

21. Divide x^ + 2 x^ y^ -\- y^ hy x'^ — xy + y\ 

22. Divide 1 — a^ by 1 + « + a^ + al 

Ans. 1 — a. 

23. Divide 10 x^ — 2Qx''y-\- 30 ^ by a; + y. 

24. Divide 7 a a;* + 21 a x^ + 14 a by a: + 1. 

Ans. 7 a a;^ + 14 a ar^ — 14 a a: -|- 14 a. 

25. Divide 27 a^ y^ — 8 a^y by 3^/^ — 2 ay. 



DEMONSTRATION OF THEOREMS. 4S 

SECTION YIII. 

DEMONSTRATION OF THEOREMS. 

57. From the principleB already established we are pre- 
pared to demoustrate the foUowiug theorems. 

THEOREM I. 
The sum of two quantities plus their difference is twice 
the greater ; and the sum of two quantities minus their dif- 
ference is twice the less. 

Let a and b represent the two quantities, and a ^ & ; their sum 
is a -\-b] their difference, a — b. 

PROOF. 

1st. {a + b) + {a — b)=a-{-b-\-a^b = 2a; 
2d. (a-\-b) — (a — b)=a + b — a+ b = 2b. 

Therefore, when the sum and difference of two quanti- 
ties are given to find the quantities, 

RULE. 

Subtract the difference from the sum, and divide the re- 
mainder by two, and ice shall have the less ; the less plus the 
difference will he the greater. 

In the following examples the sura and difference are 
given and the quantities required. 

1. 16 and 12. Ans. 14 and 2. 

2. 272 and 18. 

3. 456 and 84. 

4. Sum 2 X and difference 2 y. 

Ans. x-\-y and x — y. 

5. Sum Ix'^ + Zy and difference bx'^ — Zy. 

6. Sum 2 a — 8 ft and difference 10 a + 14 b. 

Ans. 6 a + 3 6 and — 4 a — 11 ft. 



44 ELEMENTARY ALGEBRA. 

THEOREM II. 
58. The square of the sum of two quantities is equal to 
the square of the first, plus twice the product of the two, 
plus the square of the second. 

Let a and h represent the two quantities ; their sum will be 
a-\-b. 

PROOF. 

a +h 
a +b 



a''+ ah 
+ ab + h'' 

a^-\-2ab + b'' 

According to this theorem, find the square of 

1. X -{- 1/. Ans. a:^ -j- 2 a: y -f- y • 

2. 2x + 2y. Ans. 4a:2_|-8x^/ + 4/. 

3. a:+ 1. 

4. 4: + X. 

5. 2x + St/. Ana. 4:x^+l2xy + 9f. 

6. Sa-\-b. 

THEOREM III. 
59. The square of the difference of two quantities is 
equal to the square of the first, minus twice the product of 
the two, plus the squar-e of the second. 

Let a and h represent the two quantities, and a ]> 6 ; their dif- 
ference will be a — h. 

PROOF. 

a — b 
a — b 



a^ — ab 
— ab + b^ 

a2 _ 2 a 6 + 63 



DEMONSTRATION OF THEOREMS. 45 

According to this theorem, find the square of 

1. X — y. Ans. x^ — Ixy '\- y^. 

2. 2x — 4y. 

3. X— 1. Ans. x^ — 2a:+ 1. 

4. Tx — 2. 

THEOREM IV. 

60. The product of the sum and difference of two quan- 
tities is equal to the difference of tJieir squares. 

Let a -^ b he the sum, and a — b the difference of the two 
quantities a and b. 

PROOF. 

a +b 
a —b 
a-' + ab 
--ab — b' 

According to this theorem, multiply 

1. X -\- y hy X — y. Ans. x^ — y^. 

2. 2 3:+ 1 by 2a:— 1. 

3. ar» + t/2 by x* — y^. Ans. x* — y*. 

4. 3 X + 4 by 3 a: — 4. 

6. 3xy-["4a6by3xy — 4a J. 

61 • This theorem suggests an easy method of squaring 
numbers. For, since a^ = {a — b) (a -\- b) -\- If^y 
992 = (99 — 1) (99 + 1) + 1* = 98 X 100 + 1 = 9801. 
In like manner, 

96«= 92 X 100 + 16 = 9216. 
998^ = 996 X 1000 + ^ = 996004. 
4972 = 494 X 500 + 9 = 247 X 1000 + 9 = 247009. 



46 ELEMENTARY ALGEBRA. 

In accordance with this principle find the square of 

1- 98. 4. 493. 7. 888.- 

2. 89. 5. 789. 8. 999. 

S. 45. 6. 698. 9. 1104. 

Miscellaneous Examples. 

1. Find the square of 3 a: — 6y. 

Ans. 9x^ —Mxy-\-S6f. 

2. Find the square of 4:axy -\- 7 abx. 

Ans. 16a^x^y^-[- 5Qa^bx'^i/ + 4:9an^x^. 

3. Multiply 1 x-\-l by T:r— 1. Ans. 49 x^ — 1. 

4. Required those two quantities whose sum is Bx-\-2a 
and difference x — 2 a. Ans. 2 a; and x -\- 2 a. 

6. Expand (rr^ — 4)2. 

6. Multiply 4:ab -}- S by 4aJ— -3. 

T. Find the square of 14: aH^ + 10 a?^ 2^. 

8. Find the square of 4 a — b. 

9. Multiply 10 a: 4- 2 by 10 a; — 2. 

10. Find the square ofSaa; — ^axy. 

Ans. 9a'x^ — 4.^a''x'y-\-Ua^x^f. 

11. Find the square of 2 a + 5. 

12. Find the value of (6 a + 4) (6 a — 4) (36 a^ + 16). 

Ans. 1296 a^ — 256. 

13. Find the square of 10 a^ _ 5 52 

14. Expand (3 a^a: + 4 bfy. 

Ans. 9 a* a:2 _|_ 24 ^2 j^y8 _j_ jg ^2^^«^ 

15. Find the product of a}'' + 1, «» -f 1, a^ _|_ j^ ^2 _j_ |^ 
a+l'^nda— 1. An^. a^^ — I. 

16. Find the product of a -[- i, a — 5, and a^ — 5^ 



FACTORING. 47 

SECTION IX. 

FACTORING. 

62. Factoring is the resolving a quantity into its fac- 
tors. 

63. The factors of a quantity are those integral quanti- 
ties whose continued product is the quantity. 

Note. — In usin^ the word factor we shall exclude unity. 

64* A Prime Quantity is one that is divisible without 
remainder by no integral quantity except itself and unity. 

Two quantities are mutually prime when they have no 
common factor. 

%5» The Prime Factors of a quantity are those prime 
quantities whose continued product is the quantity. 

66. The factors of a purely algebraic monomial quan- 
tity are apparent. Thus, the factors of arbxyz are 

67. Polynomials are factored by inspection, in accord- 
ance with the principles of division and the theorems of 
the preceding section. 

CASE I. 

68. When all the terms have a common factor. 
1. Find the factors of ax — ah -\- ac. 

OPERATION. As a is a factor of 

(ax — ah -\- ac) z= a {x — 6-j-c) each term it must be 

a factor of the poly- 
nomial ; and if we divide the polynomial by a, we obtain the other 
factor. Hence, 



48 ELEMENTARY ALGEBRA. 

RULE. 
Write the quotient of the poly-nomial divided by the com- 
mon factor^ in a parenthesis, with the common factor pre- 
fixed as- a coefficient. 

2. Find the factors oi ^ xy ^12 xf -\- l^ ax^ f. 

Ans. Qxy{l — l2y-\-^axy^). 

Note. — Any factor common to all the terms can be taken as well 
as 6 a; y ; 2, 3, a:, ?/, or the product of any two or more of these quan- 
tities, according to the result which is desired. In the examples 
given, let the greatest monomial factor be taken. 

3. Find the factors of a: -|- x"^. Ans. x {\ -\- x). 

4. Find the factors of 8 a^ a;^ + 12 a^ ar^ — • 4 a a:y. 

Ans. 4 a a; (2ax -\- S a^ x^ — y). 

6. Find the factors of 5 x^ y'^ + 25 ax^ — 15 x^f. 

Ans. 5 a^ (xy"^ -\- 5 ax^ — 3 y). 

6. Find the factors of 7 a a: — %hy -^-l^x^. 

T. Find the factors of 4 x^ y^ — 28 x'' y'^ — 44 ?:* y'^. 

8. Find the factors of 55 a^ c — 11 a c -\- ^^ a^ c x. 

9. Find the factors of ^^ a" x^ — 2^1 a^ x"" y\ 

10. Find the factors o^ Ibd'^ cd — ^ ah'' d^ -\-l%a^(?d\ 

CASE II. 
69. When two terms of a trinomial are perfect squares 
and positive, and the third term is equal to twice the 
product of their square roots. 

1. Find the factors oi a" + 2 ah -\-h\ 

OPERATION. Y^Q resolve this into 

a^ -\- 2 ah -\- y^ z= {a -\- h) {a -\- h) its factors at once by 

the converse of the 
principle in Theorem II. Art. 58. 



FACTORING. 49 

2. Find the factors of a' — 2 a J + ^. 

OPERATION. 'VVe resolve this into 

a* — 2 a i + 6' = (a — b) (a — b) its factors at once by 

the converse of the 
principle in Theorem IIL Art 59. Hence, 

RULE. 
Omitting the term thai is equal to twice the product of 
the square roots of the oUier two, take for each factor the 
square root of each of the other two connected by the sign 
of the term omitted. 

3. Find the factors of x* — 2a;y +5^. 

Ans. (x—y) (x — y). 

4. Find the factors of 4: a^ c^ -{- 12 acd + 9 d\ 

Ans. (2ac + 3d) (2ac-}-3d). 
6. Find the factors of 1 — 4: x z -\- 4: x^ z^. 

Ans. (1 — 2xz) (1 — 2x2). 
6. Find the factors of9a:2_g^_j_l 

Ans. (3a:— 1) (3a:— 1). 
1. Find the factors of 25 a:* + 60 a; + 36. 

8. Find the factors of 49 a^ — 14 a a: + ar». 

Ans. (la — x) (la — x). 

9. Find the factors of 16^2—16 0*^ + 4 0*. 

10. Find the factors of 12 ax + 4ar' -f 9 a^ 

11. Find the factors of 6 x + 1 + 9 x». 

CASE III. 

70. When a binomial is the difference between two 
squares. 

1. Find the factors of a^ — ^. 

OPERATIOX. We resolve this into its fac- 

q2 if^ z=z (a -[- b) (a b) ^^ ** ^"^^ ^^ ^'*® converse of 

, the principle in Theorem IV. 

Art 60. Hence, 

S D 



50 . ELEMENTAEY ALGEBRA. 

RULE. 

Take for one of the factors the sum, and for the other 
the difference, of the square roots of the terms of the hi- 
nomial. 

2. Find the factors of oiP — y^. 

Ans. (x+y) (a; — y). 

3. Find the factors of 4 a^ _ 9 b\ 

Ans. (2a + 3 52) (2 a — 3 J^). 

4. Find the factors of 16 x"^ — c^. 

5. Find the factors of a^^^c^ — oc^y^. 

6. Find the factors of 81x^ — 49/. 
1. Find the factors of 25 0^ — 4 c*. 

8. Find the factors of m^ — w^*^. 

Note. — When the exponents of each term of the residual factor 
obtained by this rule are even, this factor can be resolved again by 
the same rule. Thus, x^ — ?/* = (oi^ -\- y^) {a? — y^) ; but 7? — ^ = 
(x -f- 2^) {x — y) ; and therefore the factors of x^ — y^ are a:^ -|- ^, 
X -\-y, and x — y. 

9. Find the factors of a^ — h\ 

Ans. (a2 -f 6=2) (« + h) {a — h). 

10. Find the factors of a;^ — y'^. 

Ans. {x^ + y') (x^ + f) {x + y) (a: - y). 

11. Find the factors of a* — 1. 

12. Find the factors of 1 — x^. 

Ans. (1 + x') (1 + a:2) (1 _|_ ^) (i __ ^y 

13. Find the factors of a' — a^. 

Ans. a^{a-\- 1) (a — 1). 

14. Find three factors of x^ — x^. 

71. Any binomial consisting of the difference of the 
same powers of two quantities, or the sum of the same 
odd powers, can be factored. For . 



FACTORING. . 51 

I. T?ie difference of the same powers of tv30 quantities 
is divisible by the difference of the quantities. 

Ivct a and b represent two quantities and a ]> 6, and by actual 
division we find 

^ = -' + -^ + ^' 

and so on. 

II. The difference of the same even powers of two quan-- 
tities is divisible by the sum of the quantities. 



a -f 6 
0* — 6* 



= a — h, 



a +6 

~~^ = a^ — aH -\- a^V" — a^l^ + ah'' — ¥, 
and so on. 

It follows from the two preceding statements that 

The difference of the same even powers of two quantities 
is divisible by either the sum or the difference of the quan- 
tities. 

III. TTie sum of the same odd powers of two quantities is 
divisible by the sum of tJie quantities. 

"^^^ = a* — a«6 + a«i» — a P+ 5*, 
a -{- b ' ' ' 

^^ = a« — a* J + a*i^ — a« J^ + a*6* — a^ + 5^, 
and so on. 



52 ELEMENTARY ALGEBRA. 

1. Find the factors of oc^ — y^. 

OPERATION. 

(x» — 7/) --- (x -^ ij) = x^ + of'y + c^y^ -{- xf + y* 

By I. of this article, the difTerence of the same powers of two 
quantities is divisible by the difference of the quantities; therefore 
X — 7/ must be a factor of x^ — y^; and dividing x^ — 1/ by x — y 
gives the other factor x^ -^ x^ y -\- x^ y^ -\- x 1/ -\- y*. 

2. Find two factors of c° — d^. 

OPERATION. 

(c^ — d') -i- (c + d) =zc'-^c^d + (^d^ — c^d^ + cd^-^d^ 

By 11. the difference of the same even powers of two quantities 
is divisible by the sum of the quantities ; therefore c -\- d must be a 
factor of c® — r/"; and dividing c® — r/* by c -\- d gives the other 
factor c' — c*d-}- c' d^ — c-'^ d' -^ c d' — d\ 

3. Find the factors of m^ -\- n^. 

OPERATION. 

{m^ -|- n^) -T- {m -\- n) =:^'m^ — m^ n -\- m^'nP' — mn^ -\- n^ 

By III. the sum of the same odd powers of two quantities is 
divisible by the sum of the quantities ; therefore m-\- n must be a 
factor of m'^ -j- n^ ; and dividing irv' -\- n^ by m -\- n gives the other 

factor 771* 77l' 71 -|- 771^ f? — 771 71^ -|- 7i*. 

4. Find the factors of c^ — 7?. 

Ans. (a — x) {c? -\- a X •\- :ji?) , 

6. Find the factors of a^ -\- x^. 

Note. — In Example 2, the factors of c* — rf' there obtained are 
not the only factors; for by I. c' — d^ is divisible by c — c/; and 
dividing c* — d^ by c — d gives another factor, 

c^ -\. c' d ^ (^ d^ ■\- (? d' ■\- c d^ ■\- d'-, 
or by Art. 70, 

c« — d!« « (c« + rf») (c» — d»). 



FACTORING. 53 

But c* — c*rf-[-c»6/«--c»(i»-t-crf* — rf», 

c^-^c*d-\-c'd'-{-c'd'-{-cd*-^d\ 

are not prime quantities ; for the first can be divided by c — c?, and 
the quotient thus arising can be divided by c* ± c J -j- t/' ; the second 
can be divided by c -j- </, and the quotient thus arising will be the 
same iis after the division of the first quantity by c — </, and can 
be divided by c* ± c </ -j- </'; the third can be divided by c -f- d, and 
the fourth by c — d. Performing these divisions, by each method 
we shall find the prime factors of c* — </' to be 

c 4- J, c — (/, c' + c (/ -f- ^/», and c« — c (/ + d\ 

In finding the prime factors, it is better to apply first the princi- 
ple of Art 70 as far as possible. 

6. Find the prime factors of x^^ — y^^. 

Ans. {x+y)(x—y) (x^ — 7^y-{-x'y'^ — xf 
+ y') (x-^ + x'^y + x^f + ^y + y% 

7. Find the prime factors of «® — 1. 

Ans. (a+1) (a — 1) (a^ + a + l) (o» — a + 1). 

8. Find the prime factors of a^ — 2 a- x^ -\- x^. 

Ans. (a -{- x) (a + ^) i'^ — ^) (" — ^)- 

9. Find the prime factors of x« + 2 a:* y* + /. 

Ans. (x+y) {x + y) {j?^xy + f^) {pt^-xy + f). 

10. Find the prime factors of 1 — a*. 

Ans. (1 +a) (1 — a) (1 + a^). 

11. Find the prime factors of 8 — c*. 

Ans. (2--C) (4 + 2c + c^. 



i>4 ELEMENTARY ALGEBRA. 

SECTION X. 

GREATEST COMMON DIVISOR.* 

72. A Common Divisor of two or more quantities is any 
quantity that will divide each of them without remainder. 

73. The Greatest Common Divisor of two or more 
quantities is the greatest quantity that will divide each 
of them without remainder. 

74. To deduce a rule for finding the greatest common 
divisor of two or more quantities, we demonstrate the 
two following theorems : — 

Theorem I. A common divisor of two quantities is also 
a common divisor of the sum or the difference of any 
multiples of each. 

Let A and B be two quantities, and let d be their common di- 
visor ; d is also a common divisor of m ^ dz n B. 

Suppose A -^ d == p'^ i. e. A = dp, and mA = dmp, 
and B -^ d = q; I. e. B = d q, and n B == d n q\ 

then mA±nB = d7np±dnq = d (mp ± n q). 

That is, d is contained in m A -\- n B, m p -\- n q times, and in 
m A — n B, mp — n q times ; i. e. cZ is a common divisor of the 
sum or the difference of any multiples of A and B. 

Theorem II, The greatest common divisor of two quan- 
tities is also the greatest common divisor of the less and 
the remainder after dividing the greater by the less. 

Let A and B be two quantities, and A"^ B; 
and let the process of dividing be as appears in B) A (q 

the margin. Then, as the dividend is equal to qB 

the product of the divisor by the quotient plus ~ 

the remainder, 

A = r-^qB. (1) 

* See Prefia«e. 



GREATEST COMMON DIVISOR. 55 

And, as the remainder is equal to the dividend minus the product 
of the divisor hy the quotient, 

r= A—qB. (2) 

Therefore, according to the preceding theorem, from (1) any divisor 
of r and B must be a divisor of A ; and from (2) any divisor of A 
and jS, a divisor of r ; i. e. the divisors of A and B and B and r 
are identical, and therefore the greatest common divisor of A and 
B must also be tlie greatest common divisor of B and r. 

In the same way the greatest common divisor of B and r is the 
greatest common divisor of r and the remainder after dividing B 
by r. 

Hence, to find the greatest common divisor of any two 
quantities, 

RULE. 

Divide Oie greater by the less, and the less by the remain- 
der, and so continue till the remainder is zero; the last dir 
visor is the divisor sought. 

Note 1. — The division by each divisor should be continued until 
the remainder will contain it no longer. 

Note 2. — If the greatest common divisor of more than two quan- 
tities is required, find the greatest common divisor of two of them, 
then of this divisor and a thirds and so on ; the last divisor will 
be the divisor sought. 

Note 3. — The common divisor of x i/ and z r is a: ; x is also the 
common divisor of x and x z, or of ax y and xz\ i. e. the common 
divUor of two quantities is not changed hy rejecting or introducing 
into either any factor which contains no factor of the other. 

Note 4. — It is evident that the greatest common divisor of two 
quantities contains all the factors common to the quantities. 

CASE I. 
75» To find the greatest common divisor of monomials. 
1. Find the greatest common divisor of S a"^ li^ c d, 
WaH'c\ and 2S aH* c. 

The greatest common divisor of the coefficients found by the gen- 
eral rule is 4 ; it is evident that no higher power of a than a*, of 



66 ELEMENTARY ALGEBRA. 

h than 5', of c than itself, will divide the quantities ; and that d will 
not divide them ; therefore, the divisor sought is 4 a^ b^ c. Hence, 

RULE. 
Annex to the greatest common divisor of the coefficients 
those letters which are common to all the quantities, giving to 
each letter the least exponent it has in any of the quantities. 

2. Find the greatest common divisor of 63 a^ b'' c^ d^, 
27 a^ b' c^ and 45 a^ b"" c« d. Ans. 9 a^ h' c\ 

3. Find the greatest common divisor of ^bx'y^z^ and 
l2babxUfz\ 

4. Find the greatest common divisor of 99 a S^c^rf^a:^^ 
and 22 a" b^ c^ d^ xK Ans. II a b'^c'^d^ a^. 

5. Find the greatest common divisor of 11 x^y"^, l^si^y^, 
and 2l2bx'y^^. 

CASE II. 

76. To find the greatest common divisor of polynomials. 

1. Find the greatest common divisor of a;^ — y^ and 
x"^ — 2xy -\- y"^. 

x'-^f)x^ — 2xy-{- f(l 
x^ — y"^ 

^2xy + 2y^ 

Kejecting the factor 2 y 

X— y)x'^ — y^(x + y 
x^ — xy 

xy — y^ 

xy — y^ Ans. x — y. 

RULE. 
Arrange the terms of both quantities in the order of the 
powers of some letter^ and (lien proceed according to the 
general rule in Art. 74. 

Note 1. — If the leading term of the dividend is not divisible by 
the leading term of the divisor, it can be made so by introducing 



GREATEST COMMON DIVISOB. 67 

in the dividend a factor which contains no factor of the divisor ; or 
either quantity may be simplified by rejecting any factor which 
contains no factor of the other. (Art. 74, Note 3.) 

Note 2. — Since any quantity which will divide a will divide — a, 
and vice versa, and any quantity divisible by a is divisible by — a, 
and vice versa, therefore all the signs of either divisor or dividend, 
or of both, may be changed from -|- to — , or — to -|-» "without 
changing the common divisor. 

Note 3. — When one of the quantities is a monomial, and the other 
a polynomial, either of the given rules can be applied, although gen- 
erally the greatest common divisor will be at once apparent. 

2. Find the greatest common divisor of ax* — a^ar* — 8 a^x^ 
and 2car* — 2aca:" + 4a=^car* — 6a^cx — 20a*c. 



ax* — o'z* — Bc^x* 

DlTidlng by a Z* 

ar* — a z* — 8 a* 



2a«x» — 3a»x — 2a* 

DlTidlng by a" 

2z» — 3ax— 2a» 



2 car* — 2ac3^-\-4a*ca^ — Sc^cx — 20a*c 

Dividing by 2 C 

x* — ax^-\-2a'x' — 3a*x—l0a* (1 

a;* — gg' — 8a* 

2a'z' — 30*3: — 2a* ntBem. 

3^— ax*— 8a* 

Multiplying by 2 

2 a;* — 2 a x» — 16 a* (a:" 

2 a:* — 3 ga:* — 2 0*0:* 

a 3* -\- 2 a* x"— 16 a* 

Multiplying by 2 

2ax»-f4a»x'— 32a* (aa; 

2a3* — 3a*x* — 2a*z 

7a*x»+ 2a»z — 32a* 

Multiplying by 2 

14 a« a:* -f 4 o* a: — 64 a* (7 a« 
14a'a:*— 21a'a:— 14 a* 

25 a" ar — 50a* 25a'x — 50a* sdBem. 

DlTlding by 25 O* 

X— 2a)2a:' — 3ax— 2a'(2z-|-a 
2a:'— .4ax 

ax— 2a" 

ax — 2a* Ans. x — 2a. 



58 ELEMENTARY ALGEBRA. 

3. Find the greatest common divisor of a* — x* and 
a^ -\- a^x — ax^ — x^. Ans. a^ — x^. 

4. Find the greatest common divisor of a* — x* and 
a® — a^ x^. Ans. c^ — x^. 

5. Find the greatest common divisor o£2ax^ — a^x — a^ 
and 2 x^ -\- S a X -\- a^. 

6. Find the greatest common divisor of 6 a x — S a 
and Q ax^ -\- ax^ — 12 ax. Ans. Sax — 4a. 

v. Find the greatest common divisor of x^ — y^ and 

8. Find the greatest common divisor of 3 ic^ — 24 aj — 9 
and 2x^—16x — Q. 

9. Find the greatest common divisor of x^ — y^ and 
x^ — y^. Ans. x — y. 

10. Find the greatest common divisor of 10 x* — 20x'^y 
4-30/ and x^ + 2x^y -\- 2xy^ + yK Ans. x + y. 

11. Find the greatest common divisor of a* + "^ H~ ^^ 
+ a — 4 and a^ + 2 a« + 3 a^ + 4 a — 10. 

Ans. a — 1. 

12. Find the greatest common divisor of 1 ax^ -\- 21ax^ 
-f- 14 a and x^ -{- x^ -{- x^ — x. Ans. x^ I. 

13. Find the greatest common divisor of 21 a^y^ — 8 a^y 
and 3/ — 2 a/ 4- 3 aV — 2 a^y\ 

Ans. By"^ — 2 ay. 

14. Find the greatest common divisor of a^ -{- a — 10 
and a'^ — 16. Ans. a — 2. 

Note 5. — The greatest common divisor of polynomials can also be 
found by factoring the polynomials, and finding the product of the 
factors common to the polynomials, taking each factor the least num- 
ber of times it occurs in any of the quaptities. (Art. 74, Note 4.) 

15. Find the greatest common divisor of3a^^ — 4aar-f- 
Saxy — 4: ay and a^ x — x -\- a^y — y. 

Sax^ — 4:ax -{- Saxy — 4zay = a(x -\- y) (Sx — 4) 
a^x — X -{- a^y — yz=:(x-\-y)(a — I) (a^ -\- a -\- 1) 

Ans. so -\- y. 



LEAST COMMON MULTIPLE. 60 

SECTION XI. 

LEAST COMMON MULTIPLE. 

77. A Multiple of any quantity is a quantity that can 
be divided by it without remainder. 

78. A Common Multiple of two or more quantities is 
any quantity that can be divided by each of them with- 
out remainder. 

79. The Least Common Multiple of two or more quan- 
tities is the least quantity that can be divided by each 
of them without remainder. 

80. It is evident that a multiple of any quantity must 
contain the factors of that quantity ; and, vice versa, any 
quantity that contains the factors of another quantity is 
a multiple of it : and a common multiple of two or more 
quantities must contain the factors of these quantities ; 
and the least common multiple of two or more quantities 
must contain only the factors of these quantities. 

CASE I. 
To find the least common multiple of monomials. 

1 . Find the least common multiple of 6 a^b^c, Sc^lr^c^d, 
and 12 a* b ex. 

The least common multiple of the coefficients, found by inspection 
or the rule in Arithmetic, is 24 ; it is evident that no quantity which 
contains a power of a less than a*, of b leas than fc*, of c less than 
c\ and which does not contain d and a:, can be divided by each of 
these quantities ; therefore the multiple sought is 24 a* l^ c^ d x. 

Hence, iu the case of monomials, 



60 ELEMENTARY ALGEBRA. 

RULE. 

Annex. (0 the least common multiple of the coefficients all 
the letters which appear in the several quantities, giving to 
each letter the greatest exponent it has in any of the quan- 
tities. 

2. Find the least common multiple of 3 a* h^ c^ QaH*e d\ 
and IQahcx^. Ans. 30 a'' i^c^^^^^ 

3. Find the least common multiple of IQahx, 80 a Z»^a:^ 
and ZbaHx\ Ans. 560a^Z»^ar^ 

4. Find the least common multiple of Qa^b^, Iba'^bx^, 
and l^axy'^. Ans. ^Q a'^b^x^y^. 

6. Find the least common multiple of l^a^bc^x, 
2^ab^cx^y, and Z^aH'^xz. 

6. Find the least common multiple of Ki^xyz, 4.5 a be, 
and 25 m n. 

1. Find the least common multiple of 10 a^ by"^, 13 a^ b"^ c, 
and llan^c^ 

8. Find the least common multiple of 14 a^ b'^ c*, 20 a^ b c*, 
25a«6c», and 28 abed. 



CASE II. 

81 1 To find the least common multiple of any two 
quantities. 

Since the greatest common divisor of two quantities contains all 
the factors common to these quantities (Art. 74, Note 4) ; and since 
the least common multiple of two quantities must contain only the 
factors of these quantities (Art. 80) ; if the product of two quanti- 
ties is divided by their greatest common divisor, the quotient will 
be their least common multiple. 

Hence, to find the least common multiple of any two 
q^uautities^ 



LEAST COMMON MULTIPLE. Oft 

RULE. 
Divide one of the quantities by their greatest common di- 
xrisor, and multiply this quotient by the other quantity, and 
the product unit be tJieir least common multiple. 

Note 1. — If the least common multiple of more than two quanti- 
ties is required, find the least common multiple of two of them, 
then of this common multiple and a third, and so on ; the last com- 
mon multiple will be the multiple sought. 

Note 2. — In case the least common multiple of several monomials 
and polynomials is required, it may be better to find the least com- 
mon multiple of the monomials by the Rule in Case I., and of the 
polynomials by the Rule in Case II., and then the least common 
multiple of these two multiples by the latter Rule. 

1. Find the least common multiple of x^ — y^ and 
x' — 2xy + y\ 

OPERATION. Their greatest common 

X — y) x^ — 2xy-[-y divisor is x — y, with 

which we divide one of 

""^ y the quantities ; and mul- 

(^ ir) (^ — y)> Ans. tiplying the other quan- 

tity by this quotient, we 
have the least common multiple (x* — ^ (x — y). 

2. Find the least common multiple of 2 a* a:", ^ar^y, 
a* — X*, and a* — a' oc^. 

The least common multiple of the monomials is 4 a' x* y ; and 
the least common multiple of the polynomials is a' (a* — x*). 

The greatest common divisor of these two multiples is a' ; and 
dividing one of these multiples by a\ and multiplying the quotient 
by the other, we have 4 a? x* y (^a* — x*) as the least common mul- 
tiple. 

3. Find the least common multiple of Sd^l^, 6a^by, 
a' — 8, and a^ — 4 a + 4. 

Ans. 6 a^ b^y («» — 8) (a — 2). 



62 ELEMENTARY ALGEBRA. 

4. Find the least common multiple of 3 a;^ — 24 a; — 9 
and 2x^—16x — 6. 

(See 8th Example, Art. 76.) 

5. Find the least common multiple of a* — x^ and 

6. Find the least common multiple of a:* — 1, x^-\-2x-{-l, 
and (x — 1)^. Ans. x^ — x'^ — x^ -\- 1. 

t. Find the least common multiple of a:^ — y^ and x^ -j" y^* 

8. Find the least common multiple of a^ -j~ ^ — ^^ ^^^ 
a* — 16. 

Note 3. — The least common multiple of any quantities can also 
be found by factoring the quantities, and finding the product of all 
the factors of the quantities, taking each factor the greatest number 
of times it occurs in any of the quantities. (Art. 80.) 

9. Find the least common multiple of x^ — 2xy-{-y'^, 
x^ — y^, and [x -f- yy. 

x^—2xy-\-y'^={x — y){x — y) 

x'-y^= (x' + f) (X + y){x- y) 
(x + yy ={x + y) (x + y) 
Hence L. C. M = (a; — ?/) (x — y) (a;^ _(- y'^) (x -\-y) (x + y) 
= x^ — x^y^ — x'^,y^ + y^. 

10. Find the least common multiple of Bax^ — 4aa?-(- 
S axy — 4: ay and a^ x — x -\- a^y — y. 

(See 15th Example, Art. 76.) 

Ans. a{x -\- y) (S x — 4.) (a'' + a + 1) (a — l)= Sa'x'' — 
4:a^X -\- Ba^xy — 4a'*?/ — Sax'^ -\- 4,ax — Baxy + 4a?/. 



FRACTIONS. - 63 

SECTION XII. 

FRACTIONS. 

82. When division is expressed by writing the dividend 
over the divisor with a line between, the expression is 
called a Fraction. As a fraction, the dividend is called 
the numerator, and the divisor the denominator. 

Hence, the value of a fraction is the quotient arising 
from dividing the numerator by tlie denominator. 

X V 

Thus, - is a fraction whose numerator is z y and denominator y, 
and whose value is x. 

83. The principles upon which the operations in frac- 
tions are carried on are included in the following 

THEOREM. 
Any multiplication or division of the numerator causes a 
like change in the value of (lie fraction, and any multiplica- 
tion or division of the denominator causes an opposite change 
in tlw value of (lie fraction. 

Let — ^ be any fraction ; its value == a: y. 

Ist. Changing the numerator. 

Multiplying the numerator by y, 

which is y times the value of the given fraction. 
Dividing the numerator by y, 

— *> 

y 

which is — of the . value of the given fraction. 



64 ELEMENTARY ALGEBRA. 

2d. Changing the denominator. 

Multiplying the denominator by y, 

which is — of the value of the given fraction. 
Dividing the denominator by y, 

which is y times the value of the given fraction. 

Corollary. — Multiplying or dividing both numerator and 
denominator by the same quantity does not change the value 
of the fraction. 

For if any quantity is both multiplied and divided by the same 
quantity its value is not changed. 

^^' y ~ cy ~ I ~^' 

84i Every fraction has three signs r one for the numer- 
ator, one for the denominator, and one for the fraction 
as a whole. 

Thus, +^- 

If an even number of these signs is changed from -\- to 
— , or — to -\-, the value of the fraction is not changed ; 
but if an odd number is changed, the value of the fraction 
is changed from -\- to — , or — to -\-. 



Thus, changing an even number, 





— xy_ -\-xy_ 

-\ry —y 


taking 






1 +^y_- . jp 



+^^=+^; 



FRACTIONS. 6^r 

and changing an odd number, 

__ + ^?/ _ 1 — _5_y __, I +^y -_ __ — ^y = _ x 
4-y "^ H-y — y — y 

The various operations in fractions are presented under 
the following cases. 

CASE I. 
85t To reduce a fraction to its lowest terms. 

Note. — A fraction is in its lowest terms when its terms are mu- 
tually prime. 

1. Reduce „:— ,— ^^ to its lowest terras. 

24 a' z y" 

operation. Since dividing both terms 

16 a' :r y 4xy 2 °^ * ^"^"'^'°" ^^^ ^^« '^°^« 

24^T^ — 67y — ry quantity does not change 

its value (Art. 83, Cor.), we 
divide both terms by any factor common to them, as 4 a' ; and both 
terms of the resulting fraction by any factor common to them, as 
2xy\ or we can divide both terms of the given fraction by their 

2 
greatest common divisor* 8 a' x y ; the resulting traction — is the 

fraction sought Hence, 

RULE. 
Divide both terms of the fraction by any factor common 
to them ; then divide these quotients by any factor common to 
them ; and so proceed till the terms are mutually prime. Or, 

Divide both terms by their ffi'eatest common divisor. 

2. Reduce -~i to its lowest terms. Ans. 

x«y» xy 

972 a' x*?/* 2x 

3. Reduce T^^.— ^i to its lowest terms. Ans. ;— , — 

408 a* ar y* 3 a' y 

24 X 1/ z 2z 

4. Reduce — - — ^— to its lowest terms. Ans. — 

12a x y a 

5. Reduce ^^-^,^'^ to its lowest terms. 

51 crbxy 



66 ELEMENTARY ALGEBRA. 

6. Reduce 7^77— 7^-r^ to its lowest terms. 

1. Reduce „ , , — ^—. — ^ to its lowest terms. 
x^-^2xy -{-f 



Ans. ^ 



a'—2ab-\-b' 



8. Reduce -5 ^ ^ ,, , ,, to its lowest terms. 



9. Reduce ^-— ^ ^ — to its lowest 

2 ex* — 2acx^ -f- Aarcx' — 4 a^ c x 

terms. . aa^ 

' ' J(2^'~^fT^' 

10. Reduce — -^ — ^- — . to its lowest terms. 

gr — x* 



CASE II. 

86. To reduce fractions to equivalent fractions having 
a common denominator. 

o c 

1. Reduce j~ and j- to equivalent fractions having a 

common denominator. 

OPERATION. y^Q multiply the numerator and 

a g bx denominator of each fraction by the 

bij tt^xy denominator of the other (Art. 83, 

T ^ ,, Cor.). This must reduce them to 

c c y / 

j~ — ^r^ equivalent fractions having a common 

denominator, as the new denominator 
of each fraction is the product of the same factors. 

ORj In the second operation we find the 

a a X least common multiple, b x y, of the 

by hxy denominators by and fta:; as each de- 



c 



_ cy 



nominator is contained in this multi- 



Wx hxy pl®' 6^ch fraction can be reduced to 

a fraction with this multiple as a de- 
nominator, by multiplying its numerator and denominator by the 
quotient arising from dividing this multiple by its denominator. 
Hence, 



FRACTIONS. 6T 

RULE. 

Multiply all the denominators together for a common de- 
nominator, and multiply each numerator into the continued 
product of all the denominators, except its own, for new 
numerators. Or, 

Find the least common multiple of Oie denominators for 
the least common denominator. For new numerators, mul- 
tiply each numerator by tJie quotient arising from dividing 
this midtiple by its denominator. 



2. Reduce — » ~. and —r- to equivalent fractions 
xy ab aby ^ 

having the least common denominator. 

. ahm nxy , '^ 

Ans. -T — » ^ > and 



a b xy ab xy a bxy 

3. Reduce -— :. rr-r^* and — 7 to equivalent frac- 

15 6 10 6 c 25 act/ ^ 

tions having the least common denominator. 

. 80 a' erf 45 ad xy , 126 a; 

^^^' TbOabcd* IbO^bVd' ^ 150abcd' 

4. Reduce — » » and z—, to equivalent fractions 

m nxy b a ^ 

having the least common denominator. 

5. Reduce , . and —^t to equivalent fractions hav- 
ing the least common denominator. 

. a" — 2a6-f-6* , a*m -\- ahm 

6. Reduce _ and — ^— r to equivalent fractions 
having the least common denominator. 

7. Reduce -, v — i — » and to equivalent frac- 

3^—y- x-\-y x — y ^ 

tions having the least common denominator. 



; 


ELEMENTARY 


ALGEBRA. 




CASE 


III. 


81 


U To add fractions. 




1. 


Find the sum of - and 

OPERATION. 


c 

X 

If anvthiner 



If anything is divided into equal 

5 c h -\- c parts, a number of these parts rep- 

"^ I ^ ^ resented by 6, added to a number 

represented by c, gives h -\- c of 

these parts. In the example given, a unit is divided into x equal 

parts, and it is required to find the sum of h and c of these parts ; i. e. 

h ^, c h -\- c 

X ~^ X X 

It is evident, therefore, that fractions that have a common denom- 
inator can be added by adding their numerators. But fractions that 
do not have a common denominator can be reduced to equivalent 
fractions having a common denominator. Hence, 

RULE. 
Reduce the fractions, if necessary, to equivalent fractions 
having a common denominator; then write the sum of the 
numerators over the common denominator. 

r. A jj m X J a . bmy 4-bnx -\- any 

2. Add -, -, and -r- Ans. ' , ' 

n y ony 

3. Add -, -J, and ^. 

4. Add ^^l m. and ^^. 

^xy b ab 8 ab xy 

30 a^ b^ - f l(^a?y^-\- 35 m 
40 a b xy 



Ans. 



5. Add -—;-, -— -y, and 

3a^ ^c d 21 a c 

6. Add -— 1 — and - — — • Ans 



Y. Add 



l-|-ti I — a I —a^ 

l_|_a ,1— a . 2+2 a" 

— ^ — and ■-— i — Ans. -— ^ ^• 

1 — a \ -\-a 1 — a^ 



FRACTIONS. 

8. Add i--t^ and ^. 

9. Add ^(-;!^^-^ and ^^. 

10. Add ^-S', 5^^^ and '-^ + " . AnB. 1. 

11. Add -^^ and ^""^ 



a:* — y* a^ — 2/* 

Ans. ' + y 



7 X 

12. Add 7nx and -— -• 

lo a 

Note. — Consider m a: = — , and then proceed as before. 

. 18 a ma: 4- 7x 

A°«- — w^- 

7 X 

13. Add a? 4- y and — t-t- 

14. Add a;2 + 2xy + y*and^-^. 

Ans. ^ + ^y-^y'-y' + i. 

% — y 

CASE IV. 

88. To subtract one fraction from another. 

c b 

1. Subtract - from -• 

X X 

OPERATION. If anj-thing is divided into x 

h c b — c equal parts, a number of these 

X X X pdrts represented by c, subtracted 

from a number represented by 6, 

leaves b — c of these parts ; i. e. = • Hence, 

*^ ' XX X 

RULE. 
Reduce the fractions, if necessary, to equivalent fractions 
having a common denominator; then subtract the numerator 
of the subtrahend from that of the minuend, and write the 
result over the common denominator. 



70 ELEMENTARY ALGEBRA. 

2. Subtract -r- from r — Ans. 



4 8 c "'8 c 

3. Subtract ;r- from ;r— • 

7 X 6 ax 

4. Subtract -z-ftzs from -r — 

19 x^ Ida 

_ CI 1 ^ ^ 29 ac „ 39 a; . 273 ar* — 116 acy 

5. Subtract — -— ^ from ^; Ans. -;— ^ ^• 

14 a^ 8xy 56 xr y 

6. Subtract ^ from :; — j — -' Ans. -s -• 

1 — a 1 -\-a a^ — 1 

T. Subtract *" , from — j— ,• 
X — 1 a; -}- 1 

oot^x , ah 4-b c n ah — he 

8. Subtract „,^^jrj from ^^— j,^- 

9. Subtract r from — i-^- Ans. 



a — h ^^ a-\-h 6^ 

10. Subtract -, r from „ ' • Ans. — 



11. Subtract 16 from 



a;* _ 1 ""^" a;2 — 1 ar» -f- 1 

a;3_7 



1 4-a; 



1 fi 
Note. — Consider 16 = — , and then proceed as before. 

. oi?-^lQx — 2Z 

Ans. --j 

1 -\-x 

yZ g 

12. Subtract , from xy. 

a — h ^ 

13. Subtract x -{- b from , ; , ♦ Ans. , , , • 



CASE V. 
89i To reduce a mixed quantity to an improper fraction. 

1. Keduce x -\- - to an improper fraction. 

o 

OPERATION. As eight eighths make 

, a 8x ^^a 8a:-|-« 3- unit, there will be in 

•"S 8'8 8 X units eight times x 

. , ,, . 8a; j8a:,a 8a;-4-a ^t 

eighths; i. e. a: = — ; and y -f - = — -^ — . Hence, 



FRACTIONS. 71 

RULE. 

Multiply the integral part by the denominator of the frac- 
tion ; to the product add the numerator if tlie sign of the 
fraction is plus, and subtract it if the sign is minus, and 
under the result write the denominator. 

Note. — By a change of the language, Examples 12-14 in 
Art. 87, and 11-13 in Art. 88, become examples under this case. 
Thus, Example 12, Art. 87, might be expressed as follows: Reduce 
tnx-f- _- to an improper fraction. 

7 
2. Reduce ar* -f- 4 to an improper fraction. 

AnB. tl+^l^. 

y 

a \ X 
8. Reduce 25 a — 25 a: -I -I— to an improper frac- 

tion. 

4. Reduce a — 1 -f- . T ^^ ^^ improper fraction. 

. a" — a 
Ans. — j— -. 

6. Reduce y -| ^—^ — to an improper fraction. 

6. Reduce — ^ (a -j- 6) to an improper fraction. 

. a* — ah 
Ans. —J 

7. Reduce x — 1 J— to an improper fraction. 

Note. — It must be remembered that the sign before the dividing 
line belongs to the fraction as a whole. 

. , z»4-l a:"— i_x« — 1 —2 2 

X — 1 -, = r— ; = — J—:' or r— -' Ans. 

x-\-l ^+1 ^+1 ^-f-l 

8. Reduce x -(- 1 ^— to an improper fraction. 

o 

Ans 



1 — X 



72 ELEMENTARY ALGEBRA. 

9. Reduce x^ — 2ax -\- c^ — - — to an improper 

fraction. 

Note. — According to the same* principle an integral quantity- 
can be reduced to a fraction having any given denominator, by 
multiplying the quantity by the proposed denominator, and under the 
product writing the denominator. 

10. Reduce x -\- \ to a fraction whose denominator is 

X — 1 . * ^ — 1 

• Ans. -• 

X — 1 

11. Reduce x — 1 to a fraction whose denominator is 
a — h. 

12. Reduce 4 a a: to a fraction whose denominator is 
a^ — z. 

CASE VI. 
90. To reduce an improper fraction to an integral or 

mixed quantity. 

Cu -* -■ ^ (2 cc I 5 fl 

1. Reduce ^ to an integral or mixed quan- 

X — z a 

tity. 

OPERATION. 

* a' 

X — 2 a) x^ — 4: a X -{- 5 a^ (x — 2 a -\ ^r— 

x^ — 2 ax 

— 2ax -\- ^ c? 

— 2 a ar -(- 4 a2 



As the value of a fraction is the quotient arising from dividing the 
numerator by the denominator (Art. 82), we perform the indicated 
division. Hence, 

RULE. 

Dimde the numerator by the denominator ; if there is any 
remainder, place it over the divisor, and annex the fraction 
so formed with its proper sign to the quotient. 



FRACTIONS. 78 

2. Reduce to an integral or mixed quantity. 

Ans. a — 4 6. 

8. Reduce ""^^ "^"^^ to au integral or mixed 

quantity. 

4. Reduce g^~^^ to an integral or mixed quantity. 

^- Reduce ^^^~e^"^"^^ to an integral or mixed 

quantity. 7 

Ans. 4y-2a + ^. 

6. Reduce "^^ *o ^ integral or mixed quan- 

tity. 

>T P«^.,^^ Sax — 10 bx — 5rx 

^- deduce ^- ±- to an integral or mixed 

quantity. 

^- ^^^^^® 2a--2b to an integral or mixed 

quantity. a ^ ^ , a 



Ans. 2 a — 2 b 



a — 6 



9. Reduce ^--^ to an integral or mixed quantity. 
10. Reduce ^-— to an integral or mixed quantity 

CASE VII. 
91. To multiply a fraction by an integral quantity. 

1. Multiply ^ by c. 

OPKRAnoN. According to the theorem 

a?4-y w cx-f-cy ^° ^^ 83, multiplying the 

ab ^ ab numerator by c multiplies 

the value of the fraction c 
times. 



74 ELEMENTARY ALGEBRA. 

2. Multiply ^-i^ by a. 

OPERATION. According to the theorem 

, I in Art. 83, dividing the de- 

— — — — X 05 = — ^ — nominator by a multiplies 

the value of the fraction a 
times. Hence, 

RULE. 
Divide the denominator by the integral quantity when it 
can be done without remainder; otherwise, multiply the nu- 
merator by the integral quantity. 

« T»r li- 1 Sax -\- 4: xy , , 

3. Multiply -^P^.~ by m + n. 

Ans. ^Zn ' 

4. Multiply ^-^^j by ab. 

Note. — Any factor common to the denominator and multiplier 
may be cancelled from both before multiplying. 

^ — a ..o ^* — «v^ ^*y — ^y A 

36 + 3c X ^2^ = 6~+ c X ^ = -h^' ^^«- 

'' ^'^'''^'y ^^x ^^ ^ - 

1. Multiply y^±^ by U (x^ ^ ^). 

Ans. 2{x^ — f) (a + x). 

8. Multiply ^^^hyx—y. 

Note. — When a fraction is multiplied by a quantity equal to its 
denominator, the product is the numerator. 






Ans. 



FRACTIONS. 76 

.9. Multiply ^y^^J^ by (x - a)«. 

10. Multiply ^^-i^ by 2r» — 2 ary + y^. 
X y 

Ans. (a + h) {x — y). 

CASE VIII. 
92. To multiply an integral quantity by a fraction. 

1. Multiply ar* + 2xy + y^ by ^-. 

OPERATION. 

4(ar« + 2xy + y«)-(x + y) = 4(x + y) 

We first multiply the multiplicand by the numerator 4; but the 

multiplier is 4 -^ (x -|- if) \ and therefore this product is a: -j- y times 
too great, and this product divided by x -j- .V must be the product 
sought. 

It is evident that the result would be the same if the division were 
performed first, and the multiplication afterward. Hence, 

RULE. 
Divide the integral quantity by the denominator when it 
can be done without remainder, and muUiply the quotient 
by the numerator. Otheruxise, multiply the integral quantity 
by the numerator, and divide live product by the denom- 
inator. 

2. Multiply a» - 3 a» i + 3 a 4« - i^ by ^._//ft^y - 



Ans. Tx(a — b). 



3. Multiply a* — x* by ^ 



a'-f X* 



g 

4. Multiply 7 a* — 4xy by . « 

. 21a»— 12xy 

5. Multiply 17(ar' — y') by ^i^- 



76 ELEMENTARY ALGEBRA. 

Note. — Since the product is the same, whichever quantity is con- 
sidered as the multiplier, by considering the integral quantity as the 
multiplier, Case VIII. becomes the same as Case VII. 

CASE IX. 
93. To divide a fraction by an integral quantity. 



According to the theorem in Art. 
83, dividing the numerator by a de- 
creases the value of the fraction a 
times. 



According to the theorem in Art. 
83, multiplying the denominator by c 
decreases the value of the fraction c 
times. Hence, 

RULE. 
Divide the numerator by the integral quantity when it loill 
divide it without remainder; otherwise, multiply the denom- 
inator by the integral quantity. 

3. Divide 7-^— by a. Ans. ri— • 

46c "^ 46c 

4. Divide -y by Uf- Ans. ^yj- 

5. Divide -r- — by Q abc. 

6. Divide l^byQaJ^r^. Ans ^"^ 



1. 


Divide - by a. 




OPERATION. 




a 1 
6-^« = 6 


2. 


Divide 7- by c. 




OPERATION. 




a a 



3267 "■' -*• 32 6's' 

T. Divide ilg^J ^y 2 (« + ^) (X + y). 



Ans. 



13 {X + yy 



FRACTIONS. 77 

CASE X. 
91 • To divide an integral quantity by a fraction. 

1. Divide x by y 

OPERATiox. ^ -7- o = - ; but the divisor is not a, 

X but a-^h. Dividing by a, therefore, 

"^ a is dividing by a divisor h times too 

X hx great, and the quotient will be h times 

a^ 'a ^^^ small; therefore the quotient sought 

. X ^^ , hx -- 
18 - X = — . Hence, 
a a 



RULE. 
Divide the integral quantity by the numerator, and mul- 
tiply the quotient by the denominator. 

« T^- .J . u 3x . 16a 

2. Divide 4 a a: by — • Ans. -g-» 

3. Divide 7x2 by ^_'. Ans. '^-^' 

^ abc 3? 

4. Divide a + 6 by -. Ans. ''^"^^^ > 

5. Divide a2 + 2ax-fa:» by ^^. 

6. Divide x* — hx^ by -• 

7. Divide 2x' + 3y by ?^i^- 

8. Divide 1 by -• Ans. -• 

Note. — Hence, the reciprocal of a fraction is the fraction inverted, 

CASE XI. 
95. To multiply a fraction by a fraction. 



1. Multiply T by — 



78 ELEMENTARY ALGEBRA. 



OPERATION. We first multiply ^ by a: ; but the 

a ax multiplier is not x, but x ~- y^ there- 

h h fore the product is y times too great ; 

d X Q, X 

ax ax and -^ -^ y = r- (Art. 93) must be 

-J- -^ y -=2 — b y ^ ' 

"y the product sought. Hence, 



RULE. 

Multiply the numerators together for a new numerator, 
and the denominators for a new denominator. 

Note 1. — Common factors in the numerators and denominators 
may be cancelled before multiplication. 

Note 2. — Cases VIL and VIIL «an be included in this by writ- 
ing the integral quantity as the numerator of a fraction, with a unit 
as the denominator. 

Note 3. — Mixed quantities may be reduced to improper fractions 
before multiplying. ■ 

2. Multiply J- by ^^. ^^^--j^d^- 

3. Multiply ^4>y 1"^^. 

4. Multiply —r- by 

^ '' aoc '' mx 

5. Multiply ^Sy'-=^. Ans.^^ 

6. Multiply ""-f^ by -^^^. 

1. Multiply -,-^"t-^ by ^. 

8. Multiply ^^by ^'_f-;. 

9. Multiply - — by ^ „ o * • Ans. ^-,~y.-^' 

^ "^ 2Xy •'2a* — 8 a* 3 -f • « 



FRACTIONS. 79 

10. Multiply ,^^_, by ,,^^^._,,^^ ' 

11. Multiply YT-^-'iTy ^y T?' ^°«- ^"i^* 

12. Multiply ^by^. 

13. Multiply y + -^ by y "•' 



a-l-3/ 

Ads. -^ ^ 



a«-i^ 



14. Multiply together ^^' 7^' ^^^ V^' 

Ads. 1. 

15. Multiply together ~^» '2_/s' *°^ T" * 

16. Multiply together a + - , 5 + - ' a°d ^ — u' 

Ans. a6v4-i^ — i 5* 

if ^ by y^ 

CASE XII. 
96i To divide a fractioD by a fraction. 

1. Divide - by -7 • 
y '' h 

OPERATION. X X 

- -f- a = — (Art. 93) ; but the 

- -4- a = — ... a 

y ay divisor is not a, but - ; we have used 

. . b 

_^ y, I __ ^ a divisor h times too great, and there- 

^^ ^ fore the quotient — is 6 times too 

I. °^ 
X ox 

small, and the quotient sought is — X ^ ■=■ (Art. 91). It will 
be noticed that the denominator of the dividend is multiplied by the 
numerator of the divisor, and the numerator of the dividend by the 
denominator of the divisor. Hence, 

RULE. 
Invert the divutor, and then proceed as in multiplication of 
a fraction by a fraction. 



80 ELEMENTARY ALGEBRA. 

Note 1. — All cases in division of fractions can be brought under 
this rule, by writing integral quantities as fractions with a unit for 
the denominator. 

Note 2. — After the divisor is inverted, common factors can be 
cancelled, as in multiplication of fractions. 

Note 3. — Mixed quantities should be reduced to improper frac- 
tions before division. 

2. Divide - by — • Ans. 

c •' n cm 

x^ X* 4 

3. Divide - by -• Ans. -^— • 

4. Divide 3^ by ^^. 

. ^. ., 20^4- 2c , 3 a . 10 a«c 4- 10 c» 

6. Divide ^,3 ', — by ^-. Ans. — — ,.. ^ , ■ 



6. Divide i^^±# by -1- 



m — n 



7. Divide ^ by ^+^?. Ans. ;|^ 

8. Divide — ^—j- by — 

a — h *' 4 

3 x* ar 3 T 

9. Divide 3 . , by ■ , • Ans. 



10. Divide ^"^7/ by ^-=1^^. 

x-}- 1 '^ 4 

11. Dmde ^ by =t 

12. Divide /"'•':-!"' . by '"' + '"" + "' ■ 

Ans. 3 (m« + n«). 



13. Divide 1 + "^-^ by -4 1. 



Ans ^ <^^ + y) + ^^ +'^)' . 
c {c — x — y) 



14. Divide x + y—^ by -4-a;-|-y. 



FRACTIONS. 81 

15. Divide ^-^ + ^^ ^y r+^- 

Ans ^IL±^)* 
A^«- (i_x«j-«- 

Note. — The division of fractions is sometimes expressed by writ- 

a 

ing the divisor under the dividend. Thus, — . Such an expression 

y 
is called a Complex Fraction. A Complex Fraction can be 

reduced to a simple one by performing the division indicated. 



16. Reduce y to a simple fraction. Ans. z— • 

5 

17. Reduce ^ to a simple fraction. 

c Ans. ' — i— • 

lex — bx 

x+l 

X 1 

18. Keduce . to a simple fraction. 

19. Reduce — , . to a simple fraction. 

\—x 
^--^ 

20. Reduce ^ . ? to a simple fraction. 

Ans. — i — 

21. Reduce — ^— ^ to a simple fraction. 

a 4- 6 Ans. (x + y) (a + h). 

Note. — A Complex Fraction can also be reduced by multiplying 
its numerator and denominator by the least common multiple of 
the denominators of the fractional parts. Thus, if both terms of 
the fraction in Ex. 16 be multiplied by 5 a:, or both in Ex. 17 by 
ex, the result will be the same as above. 

4* F 



82 ELEMENTARY ALGEBRA. 



SECTION XIII. 

EQUATIONS 

OF THE FIRST DEGREE CONTAINING BUT ONE UNKNOWN 
QUANTITY. 

97i An Equation is an expression of equality between 
two quantities (Art. 9). Tliat portion of the equation 
which precedes the sign = is called the first member, and 
that which follows, the second member. 

98. The Degree of an equation containing but one un- 
known quantity is denoted by the exponent of the highest 
power of the unknown quantity in the equation. 

An equation of the first degree, or a simple equation, is 
one that contains only the first power of the unknown 
quantity. For example, 

2 a; — a a; = 27. 

An equation of the second degree, or a quadratic equation, 
is one in which the highest power of the unknown quantity 
is the second power. For example, 

x^ — ax-=zh -\- c, or ax^ — 5 = IT. 

An equation of the third degree, or a cubic equation, is one 
in which the highest power of the unknown quantity is the 
third power, and so on. 

99. The Reduction of an Equation consists in finding the 
value of the unknown quantity, and the processes involved 
depend upon the Axioms given in Art. 13. The processes 
can be best understood by considering an equation as a pair 
of scales which balance as long as an equal weight remains 
in both sides : whenever on one side any additional weight 
is put in or taken out, an equal weight must be put in or 



EQUATIONS OP THE FIRST DEGREE. 88 

taken out on the other side, in order that the equilibrium 
may remain. So, in an equation, whatever in done to one 
side muat be done to tJie oifier, in order that the equality may 
remain. 

1. If anything is added to one member, an equal quantity 
must be added to the other. 

2. If anything is subtracted from one member, an equal 
quantity must be subtracted from the other. 

3. If one member is multiplied by any quantity, the other 
member must be multiplied by an equal quantity. 

4. If one member is divided by any quantity, the other 
member must be divided by an equal quantity. 

6. If one member is involved or evolved, the other must 
be involved or evolved to the same degree. 

TRANSPOSITION. 

100. Transposition is the changing of terms from one 
member of an equation to the other, without destroying 
the equality. 

The object of transposition is to bring all the unknown 
terms into one member and all the known into the other, 
80 that the unknown may become known. 

I. Find the value of x in the equation x-|- 16 := 24. 

Subtracting 16 from the first 

OPERATION. member leaves x ; but if 1 6 is sub- 

^ ~r 16 = 24 tracted from the first member, it 

x = 24 — 16 = 8 must also be subtracted from the 

second. 

2 Find the value of x in the equation x — b = a. 

OPERATION. Adding b to the first member 

J. ft = a gives x ; but if 6 is added to the 

__ I 1 first member it must also be added 

to the second. 



84 ELEMENTARY ALGEBRA. 

3. Find the value of x in the equation 2 x = a; -j- 16. 

OPERATION. 

Subtracting x from both mem- 



2x = a: + 16 
2a: — x= 16 



bers, we have 2 a; — a; = 16, or 



x= 16. 
a: =16 

It appears from these examples that any term which dis- 
appears from one member of an equation reappears in the 
other with the opposite sign. Hence, 

RULE. 

Any term may be transposed from one member of an 
equation to the other, provided its sign is changed. 

4. Find the value of x in the equation Sx — 15 = 4x4-5. 

OPERATION. 

Sx — 15 = 4:X-\- 6 
Transposing, Sx — 4x= 5 -\~ 15 

Uniting terms, 4 a; = 20 

Dividing both members by 4, x := 5 

6. Find the value of x in 4 x -)- 46 = 5 x -|- 23. 

Note. — Reducing, we have — x = — 23. If each member of 
this equation is transposed, we shall have 23 = a: ; i. e. 23 equals 
ar, or x equals 23. Dividing both members by — 1 will give the 
same result. Hence, the signs of all the terms of an equation may 
be changed without destroying the equality. 

6. Find the value of x in It ar + It = 19 x + 13. 

Ans. x = 2, 
1. Find the value of a: in 8 a; — 14 = 13 a; — 29. 

8. Find the value of x in 5 a: + 25 = 10 a: — 25. 

Ans. X =: 10. 

9. Find the value of x in 24x — 17 = 11 x + 74. 
10. Find the value of x in 37 x — (4 + 7) = 41 x — 23. 



EQUATIONS OF THE FIRST DEGREE. 85 

CLEARING OF FRACTIONS. 
101. To clear an equation of fractions. 

1. Find the value of a: in the equation - — 2 = -+ 1. 

OPERATION. If the given equation is mul- 

X 2 -4-1 tipHed by 6, the least common 

3 6 ' multiple of 6 and 3, it will give 

2x — 12 = a:-|-6 2x — 12 = x-|-6, an equation 

X = 18 without a fractional term. Hence, 

RULE. 

Multiply each term of the equation by the least common 
multiple of the denominators. 

Note 1. — In multiplying a fractional term, divide the multiplier 
by the denominator of the fraction and multiply the numerator by 
the quotient. 

Note 2. — An equation may be cleared of fractions by multipljnng 
it first by one denominator, and the resulting equation by another, and 
so on, till all the denominators disappear; but multiplying by the 
least common multiple is generally the more expeditious method. 

"Note 3. — Before clearing effractions it is better to unite terms 
which can readily be united ; for instance, the equation in Ex. 1, 

X X 

by transposing — 2, can be written - = - -|- 3. 

o o 

Note 4. — When the sign — is before a fraction and the de- 
nominator is removed, the sign of each term that was in the nu- 
merator must be changed. 

2. Given ^ _ ^ + 25 = 33 — ^^- 

operation. 

n -oc X X ^ X — 6 

1 ransposing 25, - — I = ^ 5 — 

Multiplying by 20, 5 x — 4 x = 160 — 10 x + 60 

Transposing and uniting, 11 x = 220 

Dividing by 11, x=:20 



86 ELEMENTARY ALGEBRA. 



Note. — The sign of the numerator of — - is -|-, and must be 

o 

changed to — when the denominator is removed ; for — (-}- 4 x) 
== — 4 a:; and so the sign of each term of the numerator of the fraction 

— must be changed when the denominator 2 is removed ; for 

— (-f 10 a: — 60) = — 10 a: -|- 60. 

102i To reduce an equation of the first degree contain- 
ing but one unknown quantity, we deduce from the preced- 
ing examples- the following 

RULE. 
Clear the equation of fractions, if necessary. 

IVanspose the known terms to one member and the un- 
known to the other, and reduce each member to its simplest 
form. 

Divide both members by the coefficient of tlie unknown 
quantity. 

Note 1. — To verify an equation, we have only to substitute in 
the equation the value of the unknown quantity found by reducing 
the equation. For instance, in Ex. 2, Art. 101, by substituting 20 

for X, in ^ — f -f 25 = 33 — ^ "~ , we have 
4 5' 2 

4 5' 2 

6 _ 4 _|_ 25 = 33 — 7, 
26 = 26. 

Note 2. — When answers are not given, the work should be veri- 
fied. 

103* Since the relations between quantities in Algebra 
are often expressed in the form of a proportion, we intro- 
duce here the necessary definitions. 



EQUATIONS OF THE FIRST DEGREE. 87 

104. Ratio is the relation of one quantity to another of 
the same kind ; or, it is the quotient which arises from di- 
viding one quantity by another of the same kind. 

Ratio is indicated by writing the two quantities after 
one another with two dots between, or by expressing the 
division in the form of a fraction. Thus, the ratio of a to 
b is written, a : b, or j ; read, a is to b, or a divided by b. 

105. Proportion is an equality of ratios. Four quan- 
tities are proportional when the ratio of the first to the 
second is equal to the ratio of the third to the fourth. 

The equality of two ratios is indicated by the sign 
of equality (=) or by four dots (::). 

Q C 

Thus, a \ b =. c : dy ox a '. b '. : c : rf, or 7 = - ,; read, a to 6 

ha 

equals c to d, or a is to b s^ c is to d, or a divided by b 
equals c divided by d. 

The first and fourth terms of a proportion are called the 
extremen, and tlie second and third the means. 

106. In a proportion the product of the means is equal 
to the product of the extremes. 

Let a : b = c : d 

a c 

Clearing of fractions, ad = be 

A proportion is an equation ; and making the product 
of the means equal to the product of the extremes is 
merely clearing the equation of fractions. 

Examples. 

1. Reduce 1^ + 10 = ^^ + 13. Ans. x = 30. 

2. Reduce 17 « — 14 = 12 x — 4. Ans. x r= 2. 



88 ELEMENTARY ALGEBRA. 

3. Reduce 6 a: -— 25 + a; = 135 — 3 a: — 10. 

Ans. x-==. 15. 

4. Reduce 3a:+5 — a! = 38 — 2a:. Ans. x = 8^. 

5. Reduce ""-^ -j- ^ =, 30 — ^-i-^- Ans. x = 12. 

6. Reduce a: — 7^ = — — • Ans. a; =: lly^. 

T. Reduce ^ + ^ + ^ + ^= 154. Ans. a: = 120. 

2 o 4 o 

8. Reduce f + |= 16 + f . Ans. a: = 24. 

9. Reduce --|-a=7 \- d. 



h c 



OPERATION. 



6 ' ^ c ' 

Multiplying byJc^, chx-\- ahch:=:zh ex — hhx-\-hcdh 
Transposing, chx — b c x-\-hhxr=:h c d h — ahch 
Factoring 1st mem., (c A — hc-\-hh)xr=hc dh — a hch 
Tx-.i. 1 m ' , /. b c dk — ah ch 





"^& "J ^" 


WiXJ.V>iVXJ.U VA ^, 


ch — hc-{-hh 


10 


Reduce 


X -\- mx =: c. 


A TTS '*» — — 




1 4-m 


11. 


Reduce 


'T «-^- 


Ans. a:= ^^ 


12. 


Reduce 


1 , h 

- + - = a;. 


Ans. X = ' — 

a c 


13. 


Reduce 


a; ' X 


Ans. X = — '— 
c 


14. 


Reduce 


«=.-^+«- 


^ns. X = 9. 


15. 


Reduce 


2 3a 
= c. 

XXX 


Ans. a: = — ^ — 



EQUATIONS OF THE FIRST DEGREE. 89 



X , X , X 



16. Reduce - + - + - = 39. 

i i \ 



2 X X 

17. Reduce - - — --\-c = d. 



18. Reduce (a — 3) x + ^ = i 

/-r -r\ 

19. 



Reduce x — / 1 — ^\ =z 5. Anfl. ar = 6. 

20. Reduce 6 — ^^^tl __ a: ._ 4. 

5 

21. Reduce 2 a: ^ =18 { 

Ans. X = 9. 

ortr»j ^17 — X „ .a: — 95 

22. Reduce - =3xH --• 

23. Reduce 2 a: — ?^^ = 14 — —'^- Ans. x = 5. 

24. Reduce 6 a: + T^ - | = 9^ - ^/ + ^. 

Note. — Before clearing of fractions, transpose 7^ and unite it 

X 1 1 X 

with 9^; also transpose — -, and unite it with -5-. 



25. Reduce 4ar+?-±i = 5 + iliil 



X 

3~ 



26. Reduce ^—i — ^^ = 21 — -"ti. Ans. a: = 39. 

Jo G 

27. Reduce - 4- -r 4- - = (/. Ans. x = r — ; ; — -.' 

a ^ b ^ c be -\- ac-\- ab 

28. Reduce — ~^ — 6 = -7-? + 7. 

ftAOj ^ — 1 /» 22 — a: 3 + x . ^ 

29. Reduce — ^ — = 6 ^- Ans. x =: 7. 

O 



90 ELEMENTARY ALGEBRA. 

on r> J in I 2a: — 22 3 x — 75 . 284. — 4.x 

30. Reduce 19 + —^ — = 1 ^ 

oi T>j 4:X-\-5 5x — 5 a- 4-1 , 

31. Reduce — -^ — = — ^ 1. 

5 4 b . 

Ans. X = 5. 

32. Reduce 1^^ - '-^ = 4.-17+ '-^. 

o 4 '6 

oo r> J A ar— 12 , _ 20a: 4- 21 1 

33. Reduce 4:X [- 5 = } • 

o ' 4 4 

34. Keduce = ~ » Ans. a; r=: 



a; c m ' hm-\-cd 

35. Reduce 7 — — = 1 — 3 « c. 

oa x> A 5a:+3 3a:-j-15 . , 6a:4-10 

36. Reduce — ^ h 6 ,' = 4 -] \ 

2 ' 4 '4 

OK -o J o 3a — 19 _ 23— a: , 5.r— 38 , -^ 
31. Reduce Zx 8 =: — — - 4 \- 10. 

z 4 o 

Ans. X = 19» 

«« _,, 13 — 3x 3a:4-2 ^ „ ,8a:— 13 

38. Reduce — ,^ J— - = T — 6 :r ^ 

10 5 '0 

on T> A 4.(x—7) 3(a:4-l) 7.t— 17 a; 

39. Reduce ^ + 11 = —lo ^ 21 ' 

,^ _., 4.r — 6,„ 19— 4a: 5.r — 6, 7 a: + 8 

40. Reduce a: 1-3=:— ^ ^—] ^. 

41. Reduce ^ + ^ + ^ + ^ = m. 

,^_., 7x4-5, 6a: — 30 ,, 

42. Reduce - J— + ~ — = x + I. 

7 ' 7 X — 7 ' 

Note. — Multiply by 7, transpose, and unite. 

43. Reduce 2 (3 4- a:) : 6 a: — 9 = 2 : 3. Ans. x= 6. 

44. Reduce l + f: ^^^J-, - H -^ 

45. Reduce b : c -\- d = - : n. 

X. ' a: 



EQUATIONS OF THE FIRST DEGREE. ftl 



PROBLEMS 

PRODUCING EQUATIONS OF THE FIRST DEGREE CON- 
TAINING BUT ONE UNKNOWN QUANTITY. 

107. The problems given in this Section must either con- 
tain but one unknown quantity, or the unknown quanti- 
ties must be so related to one another that if one be- 
comes known the others also become known. 

108. With "beginners the chief difficulty in solving a 
problem is in translating the statements or conditions 
of the problem from common to algebraic language ; i. e. 
in preparing the data, and forming an equation in accord- 
ance with the given conditions. 

1. If three times a certain number is added to one half 
and one third of itself, the sum is 115. What is the 
number ? 

SOLUTION. 

Let X =. the number ; 

then by the conditions of the problem. 

Clearing of fractions, 18x + 3a; + 2a: = 690 
Uniting terms, 23 ar = 690 

Dividing by 23, ar=: 30 

VERIFICATION. 

3X30 + f + ^»=U5 

115 = 116 

In this problem there is but one unknown quantity, which we rep- 
resent by X. 

2. There are three numbers of which the first is 6 more 
than the second, and 11 less than the third ; and their sum 
is 101. What are the numbers? 



92 ELEMENTARY ALGEBRA. 

SOLUTION. 

Let X r= the first, In this problem 

then X — 6 = the second, there are three un- 

and a: + 11 = the third. known quantities; 

Their sum,3x+ 6 = loT 1^"^*^"^ "^" ^^ ^^ 

lated to one an- 

other that, if any 

x= 32, the first, one becomes known, 

X— 6 = 26, the second, t^g other two will 

a? + 11 = 43, the third. be known. 

VERIFICATION. 

32 + 26 + 43 = 101 
101 = 101 

From these examples we deduce the following 



GENERAL RULE. 
Let X [or some one of the latter letters of the alphabet) 
represent the unknown quantity ; or, if there is more than one 
unknown quantity, let x represent one, and find the others by 
expressing in algebraic form their given relations to the one 
represented by x. 

With tJw data thus prepared form an equation in accord- 
ance with the conditions given in the problem. 

Solve the equation. 

The three steps may be briefly expressed thus : — 
1 St. Preparing the Data ; 
2d. Forming the Equation ; 
3d. Solving the Equation. 

3. The sum of three numbers is 960 ; the first is one 
half of the second and one third of the third. What are 
the numbers ? Ans. 160, 320, and 480. 

4. Find two numbers whose difference is 18 and whose 
sum 112. Ans. 47 and Qb, 



EQUATIONS OF THE FIBST DEGREE. 98 

5. A man being asked how much he gave for his horse 
said, that if he had given $ TO more than three times as 
much as it cost, he would have given S445. How much 
did his horse cost him ? 

6. A man being asked how many sheep he had, replied 
that if he had as many more, and two thirds as many, 
and three fifths as many, he should have 8 more than 
three times as many as he had. IIow many sheep had he ? 

*?. Divide $675 between A and B in such a manner 
that B may have two thirds as much as A. 

Ans. A^s share, $ 345 ; 
B^s " $230. 

8. A father divided his estate among his three children 
80 that the eldest had $ 1440 less than one half of the 
whole, the second $500 more than one third of the 
whole, and the youngest $ 250 more than one fourth 
of the whole. What was the value of the estate ? 

SOLUTION. 

Let X •=. whole estate. 

Then ^ — 1440 = share of the eldest, 

1+ 500= " " " second, 

1+ 250= " '* " youngest, 



13 X 

Their sum — 690 =: x, whole estate. 

i= 690 

X = 8280, whole estate. 

9. A gentleman meeting five poor persons, distributed 
$7.50, giving to the second twice, to the third three times, 
to the fourth four times, and to the fifth five times as much 
as to the first How much did he give to each ? 



94 ELEMENTARY ALGEBRA. 

10. Divide 195 into two such parts that the greater di- 
vided by 3 shall be equal to the less divided by 2. 

Note. — To avoid fractions, let 3 a: = the greater and 2 a; = the less. 

Ans. 477 and 318. 

11. Divide a into two such parts that the greater di- 
vided by h shall be equal to the less divided by c. 

SOLUTION. 

Let X = the greater, 

then a — x = the less. 

. , ' X a — X 

And T = 

b c 

Clearing of fractions, cx^= ah — hx 
Transposing, hx -\- ex =. ab 

Dividing by 5 -f- c, x-= r-^-^ the greater, 

ah ac , , , 

a — x=:a — f—. — = 5— j — , the less. 
b-\'C b-\-c 

12. What number is that which, if multiplied by 7, and 
the product increased by eleven times the number, and 
this sum divided by 9, will give the quotient 6 ? 

13. If to a certain number 55 is added, and the sum 
divided by 9, the quotient will be 5 less than one fifth 
of the number. What is the number? Ans. 125. 

14. As A and B are talking of their ages, A says to B, 
*' If one third, one fourth, and seven twelfths of my age 
are added to my age, the sum will be 8 more than twice 
my age.^' What was A's age ? 

15. A farmer having bought a horse kept him six weeks 
at an expense of $20, and then sold him for four fifths of 
the original cost, losing thereby $ 50. How much did he 
pay for the horse? Ans. $150. 

16. A man left $ 18204, to be divided among his widow, 
three sons, and two daughters, in such a manner that the 
widow should have twice as much as a son, and each son 
as much as both daughters. What was the share of each ? 



EQUATIONS OF THE FIRST DEGREE. 95 

IT. If a certain number is divided by 9, the sum of the 
divisor, dividend, and quotient will be 89. What is the 
number? Ans. 72. 

18. If a certain quantity is divided by a, the sum of the 
divisor, dividend, and quotient will be b. What is the 
quantity ? 

19. Verify the answer to the preceding problem. 

20. A farmer mixed together corn, barley, and oats. In 
all there were 80 bushels, and the mixture contained two 
thirds as much corn as barley and one fifth as much bar- 
ley as oats. How many bushels of each were there ? 

21. Three men, A, B, and C, built 572 rods of fence. A 
built 8 rods per day, B 7, and C 5. A worked one half 
as many days as B, and B one third as many as C. How 
many days did each work ? 

22. What number is as much greater than 340 as its 
third part is greater than 34 ? Ans. 459. 

23. A man meeting some beggars gave 3 cents to each, 
and had 4 cents left. If he had undertaken to give 5 cents 
to each, he would have needed 6 cents to complete the dis- 
tribution. How many beggars were there, and how much 
money did he have ? 

SOLUTION. 

Let X = the number of beggars ; 

then, according to the first statement, 

3 X + 4 = the number of cents he had, 
and, according to the second statement, 

bx — 6 = the number of cents he had. 
Therefore, 6a: — 6 = 3x + 4 

2a:=10 
ar = 5, the number of beggars, 
and 8x -[- 4 = 19, the number of cents he had. 



96 ELEMENTARY ALGEBRA. 

24. A boy wishing to distribute all his money among 
his companions gave to each 2 cents, and had 3 cents 
left ; therefore, collecting it again, he began to give 3 
cents to each, but found that in this case there was one 
who had received none, and another who had only 2 
cents. How many companions, and how much money 
had he? Ans. 7 companions, and IT cents. 

25. What two numbers whose difference is 35 are to 
each other as 4 : 5 ? 

26. A man being asked the hour, answered that three 
times the number of hours before noon was equal to three 
fifths of the number since midnight. What was the time 
of day ? 

SOLUTION 

Let X = the number of hours since midnight, i. e. the time ; 
then 12 — x = the number of hours before noon. 

Then . 36— 3^:=?^ 

5 

Clearing of fractions, 180 — 15 x =z^x 
Whence 18x = 180 

X =z 10. Ans. 10 o'clock. 

2Y. A gains in trade $ 300 ; B gains one half as much 
as A, plus one third as much as C ; and C gains as much 
as A and B. What is the gain of B and C ? 

Ans. B's, $375; C's, $675. 

28. What number is to 28 increased by one third of 
the number as 2 : 3 ? Ans, 24. 

29. What number is that whose fifth part exceeds its 
sixth b}^ 15 ? 

30. Divide $3740 into two parts which shall be in the 
ratio of 10 : 7. 

31. Divide a into two parts which shall be in the ratio 

of6:c, . ah , ac 

Ans. i— 1 — and 7—-, — 
h-\- c -\-c 



EQUATIONS OF THE FIRST DEGREE. 97 

32. What number ia that the sum of whose fourth part, 
fifth part, and sixth part is 37 ? 

33. What quantity is that the sum of whose third part, 

fifth part, and seventh part is a ? » 105 a 

-ft-ns. ^^ 

34. A farmer sold IT bushels of oats at a certain price, 
and afterward 12 bushels at the same rate ; the second 
time he received 55 shillings less than the first. What 
was the price per bushel ? 

35. A certain number consists of two figures whose 
sum is 9 ; and if 27 is added to the number, the order of 
the figures will be inverted. What is the number ? 

SOLUTION. 

liCt X =z the left-hand figure ; 

then 9 — x = the right-hand figure. 

As figures increase from right to left in a tenfold ratio, 
10 x + (9 — x) = 9 a: -f- 9 = the number ; 
and when the order of the figures is inverted, 
10 (9 — ar) -f- a: = 90 — 9 a: := the resulting nu mber. 
Therefore 9x+9 + 27 = 90— -9a:. 
Or 18 a: = 54 

Whence ar = 3, the left-hand figure, 

and 9 — x = 6, the right-hand figure. 

Ans. 36. 

36. A certain number consists of three figures whose 
sum is 6, and the middle figure is double the left-hand 
figure ; and if 198 is added to the number, the order of 
the figures will be inverted. What is the number ? 

Ans. 123. 

37. Two men 90 miles apart travel towards each other 
till they meet. The first travels 5 miles an hour and the 
second 4. IIow many miles does each travel before they 
meet? 

6 o 



98 ELEMENTARY ALGEBRA. 

38. A man hired six laborers, to the first of whom he 
paid t5 cents a week more than to the second ; to the 
second, 80 cents more than to the third ; to the third, 60 
cents more than to the fourth ; to the fourth, 50 cents 
more than to the fifth ; to the fifth, 40 cents more than 
to the sixth; and to all he paid $68.15 a week. What 
did he pay to each a week ? 

89. What number is that to which if 20 is added two 
thirds of the sum will be 80 ? 

40. What number is that to which if a is added - of 

c 

the sum will be c?? \^„ ^^ ^ 

Ans. -T a. 



41. A man spent one fourth of his life in Ireland, one 
fifth in England, and the rest, which was 33 years, in 
the United States. To what age did he live? 

42. A post is one fifth in the mud, two sevenths in 
the water, and 18 feet above the water. How long is 

the post ? 

43. What number is that whose half is as much less 
than 40 as three limes the number is greater than 156? 

Ans. 56. 

44. Two workmen received the same sum for their la- 
bor ; but if one had received $15 less and the other $15 
more, one would have received just four times as much 
as the other. What did each receive ? 

45. Of the trees on a certain lot of land five sevenths 
are oak, one fifth are chestnut, and there are 32 less wal- 
nut trees than chestnut. How many trees are there ? 

46. Divide 474 into two parts such that, if the greater 
part is divided by 7 and the less by 3, the first quo- 
tient shall be greater than the second by 12. 

Ans. 357 and 117. 



EQUATIONS OF THE FIRST DEGREE. 99 

4Y. Two persons, A and B, have each an annual income 
of S1500. A spends every year $400 more than B, and 
at the end of five years the amount of their savings is 
$6000. What does each spend annually? 

Ans. A $1100, and B $700. 

48. In a skirmish the number of men captured was 41 
more, and the number killed 26 less than the number 
wounded ; 45 men ran away ; and the whole number en- 
gaged was four times the number wounded. How many 
men belonged to the skirmishing party ? Ans. 240. 

49. A and B have the same salary. A runs into debt 
every year a sum equal to one sixth of his salary, while 
B spends only three fourths of his ; at the end of five 
years B has saved $1000 more than enough to pay A's 
debt What is the salary of each ? Ans. $ 2400. 

60. A man lived single one third of his life : after hav- 
ing been married two years more than one eighth of his 
life, he had a daughter who died ten years after him, 
and whose age at her death was one year less than two 
thirds the age of her father at his death. What was the 
father's age at his death ? 

SOLUTION. 

Let X =: his age ; 

then - = his age at marriage, 

o 

^ + f + 2 = his age at daughter's birth, 

8 o 

and ar — ( 1 4- ^ -|- 2 ) = her age at his death. 

Then x-f-|-2+10 = '^-l 

Transposing and uniting, — - = — 9 

X = 72, the father's age. 



100 ELEMENTARY ALGEBRA. 

51. Divide $ 864 among three persons so that A shall have 
as much as B and C together, and B $5 as often as C $ 11. 

52. A father and sou are aged respectively 32 and 8. 
How long vi^ill it be before the son will be just one half 
the age of the father ? 

53. A man's age was to that of his wife at the time 
of their marriage as 4:3, and seven years after, their 
ages were as 5:4. What was the age of each at the 
time of their marriage ? 

54. One fifth of a certain number minus one fourth of 
a number 20 less is 2. What is the number? Ans. 60. 

55. There are two numbers which are to each other as 
J : ^ ; but if 9 is added to each, they will be as i : ^. 
What are the numbers ? Ans. 9 and 6. 

56. A person having spent $ 150 more than one third 
of his income had $ 50 more than one half of it left. 
What was his income ? 

5T. A merchant sold from a piece of cloth a number 
of yards, such that the number sold was to the number 
left as 4 : 5 ; then he cut off for his own use 15 yards, 
and found that the number of yards left in the piece was 
to the number sold as 1:2. How many yards did the 
piece originally contain ? Ans. 45. 

58. Four places. A, B, C, and D, are in a straight line, 
and the distance from A to D is 126 miles. The distance 
from A to B is to the distance from B to C as 3 : 4, and 
one third the distance from A to B added to three fourths 
the distance from B to C is twice the distance from to 
D. What is the distance from A to B, from B to C, and 
from C to D ? 

59. A laborer was hired for 40 days ; for each day he 
wrought he was to receive $2.50, and for each day he 
was idle he was to forfeit $1.25. At the end of the time 
he received $58.75. How many days did he work? 

Ans. 29. 



A e.^,- 



EQfUATIONS OF THE HBST DEGREE. 101 

60. A cask which held 44 gallons was filled with a 
mixture of brandy, wine, and water. 'T^crif) 'were '19: gal- 
lons more than one half as muct ,wine as brandy, ^nd , as 
much water as brandy and wJDje/'-Hpw* ulai^y*' g&>)[^V8' 
were there of each ? 

61. Two persons, A and B, travelling each with S80, 
meet with robbers who take from A $ 5 more than twice 
what tliey take from B; then B finds he has $26 more 
than twice what A has. IIow much is taken from each ? 

Ans. From A, $69 ; from B, $32. 

62. Four persons, A, B, C, and D, entered into part- 
nership with a capital of $84810; of which B put in 
twice as much as A, C as much as A and B, and D as 
much as A, B, and C. How much did each put in ? 

63. In three cities, A, B, and C, 1188 soldiers are to 
be raised. The number of enrolled men in A is to that 
in B as 3 : 5 ; and the number in B to that in C as 8 : T. 
How many soldiers dught each city to furnish ? 

Ans. A, 288 ; B, 480 ; C, 420. 

64. Divide $65 among five boys, so that the fourth 
may have $2 more than the fiftli and $3 less than the 
third, and the second $4 more than the third and $5 
less than the first. 

65. A merchant bought two pieces of cloth, one at the 
rate of $ 5 for 7 yards, and the other $ 2 for 3 yards ; 
the second piece contained as many times 3 yards as the 
first times 4 yards. He sold each piece at the rate of 
$6 for 7 yards, and gained $24 by the bargain. How 
many yards were there in each piece ? 

Ans. First, 84 ; second, 63. 

66. A drover had the same number of cows and sheep. 
Having sold 17 cows and one third of his sheep, he finds 
he has three and a half times as many sheep as cows left. 
How many of each did he have at first? 



102 ELEMENTARY ALGEBRA. 

67 A flour dealer sold one fourth of all the flour he 
had aod one foilrth'. of: a barrel ; afterward he sold one 
third. c>f wli^t he had left .and one thh-d of a barrel; and 
ttlen. ofte Jigjf of tl^Q remainder and one half of a barrel; 
and had 15 barrels left. How many had he at first ? 

SOLUTION. 

Let X = number at first; 

3 X 1 
then — — - = number after first sale, 

2 /3 X 1\ 1 a; 1 

g{~l ;) — 3^^^2 — 2^^ number after second sale, 

and 2 §2 — 2/ — 2^^4 — 4^^ number after third sale. 

Then ^ -^ ? =:: 15 

4 4 

Clearing of fractions, x — 3 = 60 

Whence x = 63, number at first. 

« 

68. A merchant bought a barrel of oil for $50; at the 
same rate per gallon as he paid, he sold to one man 15 
gallons ; then to another at the same rate two fifths of 
the remainder for $ 14. IIow many gallons did he buy 
in the barrel ? 

69. Two pieces of cloth of the same length but dif- 
ferent prices per yard were sold, one for $5 and the 
other for St. 50. If there had been 6 more yards in 
each, at the same rate per yard as before, they would 
have come to $ 15.41^^. How many yards were there in 
each? Ans. 21. 

10. A and B began trade with equal sums of money. 
The first year A lost one third of his money, and B 
gained $ 150. The second year A doubled what he had 
at the end of the first year, and B lost $150, when the 
two had again an equal sum. What did each have at 
first ? 



EQUATIONS OF THE FIRST DEGREE. 108 

11. A man distributed among his laborers $2.50 apiece, 
and had $25 left. If he had given each $3 as long as 
his money lasted, three would liave received nothing. 
How many laborers were there, and how much money 
did he have? Ans. 68 laborers, and S195. 

72. A man who owned two horses bought a saddle for 
S35. When the saddle was put on one horse, their value 
together was double the value of the other horse ; but 
when the saddle was put on the other horse, their value 
together was four fifths of the value of the first horse* 
What was the value of each horse ? 

T3. From a cask two thirds full 18 gallons were taken, 
when it was found to be five ninths full. How many 
gallons will the cask hold ? 

H. A farmer had two flocks of sheep, and sold one 
flock for $60. Now a sheep of the flock sold was worth 
4 of those left, and the whole value of those left was $8 
more than the price of 8 sheep of those sold, and the flock 
left contained' 40 sheep. How many sheep did the farmer 
sell, and what was the value of a sheep of each flock ? 
Ans. Number sold, 15; value, $4 and $1. 

75. A man has seven sons with 2 years between the 
ages of any two successive ones, and the sum of all their 
ages is ten times the age of the youngest. What is the 
age of each ? 

76. Divide 75 into two parts such that the greater in- 
creased by 9 shall be to the less diminished by 4 as 3 : 1. 

77. Divide a into two parts such that the greater in- 
creased by b shall be to the less diminished by c as m : n. 

78. What two numbers are as 3:4, while if 8 be 
added to each the sums will be as 5 : 6 ? 

79. Divide 127 into two parts, such that the difference 
between the greater and 130 shall be equal to five times 
the difference between the less and 63. 



104 ELEMENTARY ALGEBRA. 



SECTION XIV. 

EQUATIONS 

OF THE FIRST DEGREE CONTAINING TWO UNKNOWN 
QUANTITIES. 

109. Independent Equations are such as cannot be de- 
rived from one another, or reduced to the same form. 

Thus, a; + y= 10, I + 1=5, and 4a:+32^=:40— y 
are not independent equations, since any one of the three 
can be derived from any other one ; or they can all be 
reduced to the form x -\- y = 10. But a; -j- y = 10 and 
4:X = 1/ are independent equations. 

110. To find the value of several unknown quantities, 
there must be as many independent equations in which 
the unknown quantities occur as there are unknown 
quantities. 

From the equation x -}- i/ = 10 we cannot determine the value 
of either a: or y in known terms. If y is transposed, we have 
x= 10 — y; but since y is unknown, we have not determined the 
value of X. We may suppose y equal to any number whatever, 
and then x would equal the remainder obtained by subtracting y 
from 10. It is only required by the equation that the sum of two 
numbers shall equal 1 ; but there is an infinite number of pairs 
of numbers whose sum is equal to 10. But if we have also the 
equation 4x = y, vre may put this value of y In the first equation, 
X -{- y = 10, and obtain a: -j- 4 a: = 10, or a; = 2 ; then 4 a: = 8 = y, 
and we have the value of each of the unknown quantities. 



ELIMINATION. 

Ill, Elimination is the method of deriving from the 
given equations a new equation, or equations, containing 
one less unknown quantity. The unknown quantity thus 
excluded is said to be eliminated. 



EQUATIONS OF THE FIRST DEGREE. 105 

There are three methods of elimination : — 
I. By substitution. 
II. By comparison. 

111. By combination. 

CASE I. 

112, Elimination by substitution. 

1. Given |*^ + ^y = 2^l, to find a;and y. 

OPERATION. 

4a: + 6y = 23 (1) 6a: + 4y= 22 (2) 

(3) 6x + 4(?i^)= 22 (4) 

25X + 92 — 16aj = 110 (5) 

?^ = 3 (7) x= 2 (6) 



23 — 4 z 

y = — s — 



Transposing 4 x in (1) and dividing by 5, we have (8), which 
gives an expression for the value of y. Substituting this value of 
y in (2), we have (4), which contains but one unknown quantity ; 
i. e. y has been eliminated. Reducing (4) we obtain (6), or 
a: == 2. Substituting this value of x in (3), we obtain (7), or 
y = 3. Hence, 

RULE. 

Find an expression for the value of one of the unknown 
quantities in one of the equations, and substitute this value 
for the same unknown quantify in the other equation. 

Note. — After eliminating, the resulting equation is reduced by 
the rule in Art. 102. The value of the unknown quantity thus 
found must be substituted in one of the equations containing the 
two unknown quantities, and this reduced by the rule in Art. 102. 

Find the values of x and y in the following equations : — 
2. Given 1^ + !'= in. Ans. j- = l«- 



106 ELEMENTARY ALGEBRA. 



- + ^=12 



3. Given ^ - - L. Ans. j^=^^• 

5 + ^ - . J (5^ = 15. 



4. Given 



5. Given 



(2. -I- 3 = 0) ^^^^ ( 

(a;+y — 29 = 0) ^ 

■J 2 "^ 3 36 >• . 
(2a7+ 32/ = 2 ) 



x= 1. 
^ = 22. 



6. Given ] 2~3-" ^L 
(3a: — 3^ = 16) 

CASE II. 
113« Elimination by comparison. 

1. Given {2rJy = 2?}' *^ ^^^ ^ ^^^ y- 

OPERATION. 

a: — 2y = 6 (1) 2a: — y = 27 (2) 

a: = 6 + 2s^ (3) ^_ 27 + y (^^ 



«l/ — 


" . 2 


6 + 2y=^' + ^ 


(5) 


12 + i!, = 21+y 


(6) 


P= 6 


0) 


a;= 6 + 10 = 16 


(8) 



Finding an expression for the value of x from both (1) and (2), 
we have (3) and (4). Placing these two values of x equal to 
each other (Art. 13, Ax. 8), we form (5), which contains but one 
unknown quantity. Reducing (5) we obtain (7), or* ?/ = 5. Sub- 
stituting this value of y in (3), we have (8), or a: = 16. Hence, 



EQUATIONS OF THE FIRST DEGREE. 
RULE. 



107 



Find an expression for ihe value of the same unknovm 
quantity from each equation, and put these expressions equal 
to each other. 

By this method of elimination find the values of x and y 
in the following equations : — 



{. Given -l^^""! — ^[.. 



3. Given 



+ y = 12 



= 8 



4. Given |3a: + 5y = 2| 



5. Given 




(3(x-y)-9 = 0) 

7. Given i6*-5y=17|. 
( 2a; — v = 13> 



8. Given ■ 



:-!=« 






Ana. 



( V = 3. 



Ans 



( V = 



= 24. 



Ans. 



Ans. 



(a: = 2. 



108 



ELEMENTARY ALGEBRA. 



CASE III. 
114i Elimination by combination. 

1. Given ■< ^ ^ ?- , to find x and y. 





OPERATIOS 


. 




Zx — 2y— 1 


(1) 


2a; — 3y = 3 


(2) 


Qx — iy— 14 


(3) 






6 a; — 9.y= 9 


(4) 






by= 5 


(5) 


2a; — 3 =3 


0) 


y= 1 


(6) 


a;=:3 


(8) 



If we multiply (1) by 2, and (2) by 3, we have (3) and (4), 
fn which the coefficients of x are equal; subtracting (4) from (3), 
we have (5), which contains but one unknown quantity. Redu- 
cing (5), we have (6), or y = 1 ; substituting this value of y in 
(2), we obtain (7), which reduced gives (8), or a; = 3. 



2. Given - 


-X y __ 
2 4 

13 "^ 2 


6 
12 


- , to find X and y. 




• 


DPI 


SRATION. 






!-!= ' w 






.-1=12 


(2) 
(3) 


9-1= 6 (6) 


'i-^^ 


(4) 


y = 


12 (1) 






a: = 18 


(5) 



If we multiply (1) by 2, we have (3), an equation in which y has 
the same coefficient as in (2) ; since the signs of y are different in 
(2) and (3), if we add these two equations together, we have 
(4), which contains but one unknown quantity. Reducing (4), we 
have (5), or a: = 18. Substituting this value of x in (1), we have 
(6), which reduced gives (7), or 3/ =-12. Hence, 



EQUATIONS OF THE FIRST DEGREE. 
RULE. 



109 



Multiply or divide the equations so that the coefficients of 
the quantity to he eliminated shall become equal ; then, if the 
signs of this quantity are alike in both, subtract one equa- 
tion from the other ; if unlike, add the two equations to- 
getlier. 

Note. — The least multiplier for each equation will be that which 
will make the coefficient of the quantity to be eliminated the least 
common multiple of the two coefficients of this quantity in the 
given equations.' It is always best to eliminate that quantity whose 
coefficients can most easily be made equal. 

By this method of elimination find the values of x and y 
in the following equations : — 



3. Given |^- + 3y = 33> ^^ jx 



x=3. 
4. 



4. Given 



f 8a: + 6t/ 
(lOx — 3y 



}■ 



Ans 






6. Given ■ 



19 • 6 



y __ 



12 
5 



Ans. 1^ = 27. 
\y = 12. 



6. Given 



fx-f y 



-^— 1 =0 



1. Given 



2 
T X 






110 ' ELEMENTARY ALGEBRA. 

115* Find the values of x and y in the following 

Examples. 

Note. — Which of the three methods of elimination should be 
used depends upon the relations of the coefficients to each other. 
That one which will ehminate the quantity desired with the least 
work is the best. 



1. Given P- + 32/ = 25|. 
\'!x + 2t/ = 2Si 

(5x— y= 0> 



2. Given 



3. Given 



4. Given 



2 

x—2y 
3 



— y=6 



(x—l , 17 

( 2x— .4y==17 



Ans. y ' 



Ans 



Ans. 






= 3. 
15. 



fx = ll. 

ly= 1. 
(. = 9. 



5. Given 



2a;-{-3y 1 a;-|-22/ + 3 

Z "~ 3 ~ 2 

^__2.-2y^3 



Ans, 



(a; = ll. 

(2^= 5. 



6. Given 



Y. Given 



7+3 -^ 
L 8 + 16 — "^^r 

—3 3— - ^^ 

^ — y 3: + y ___ J. 



EQUATIONS OF THE FIRST DEGREE. 



Ill 



8. Given 



2x_, 5y 8^ 

T ' T 15 
3 7 



Ans 






9. Given 



10. Given 



5x + 



54 



+ 5 y = 102 



ar + y I y^ '^y — ^ 

4 •"3'" 2 



11. Given \ 4 + 3 "= ^ [ • 
( 4 a; — 3 V = 25 ; 



Ans 



( v = 



= 10. 

20. 



m 



\ ^-l-: 



= 48. 

28. 



(.y = — 3. 



12. Given 



^ — .V _o 



Ans 






1 — 



13. Given 






5 J 



Adb, 






10. 
20. 



14. Given 



^-±y = y^2 



4y — 4 
2 



15. Given 






112 

16. Given 



ELEMENTARY ALGEBRA. 



3 — ^ 3J 



IT. Given 



r ^+y _^ 1 

"X" — ^~"3 



18. Given ^ 



2y 

3a: 



a: + 3 



4 

7 — y 

3 



6 
14 



19. Given < 



4a;— 7y 

5 
3rr + y 



X — 4 



= y + 3 



PROBLEMS 

PRODUCING EQUATIONS OF THE FIRST DEGREE CON- 
TAINING TWO UNKNOWN QUANTITIES. 

116t Many of the problems given in Section XIII. con- 
tain two or more unknown quantities ; but in every case 
these are so related to each other that, if one becomes 
known, the others become known also ; and therefore 
the problems can be solved by the use of a single let- 
ter. But many problems, on account of the complicated 
conditions, cannot be performed by the use of a single 
letter. No problem can be solved unless the conditions 
given are suflScient to form as many independent equa- 
tions as there are unknown quantities. 

1. A grocer sold to one man Y apples and 5 pears for 
41 cents ; to another at the same rate 11 apples and 3 
pears for 45 cents. What was the price of each ? 



EQUATIONS OF THE FIRST DEGREE. 113 

SOLUTION. 

Let X = the price of an apple, 
and y =z " " "a pear. 

Then, by the conditions. 



'Jx + 5y = 41 


(1) 


and Hx-f- 3y= 45 


(2; 






55x + 15ff = 225 


(3) 






21a;+15y=123 


(4) 


2l + 5yr=41 


(•?) 


34x=102 


(5) 


y= 4 


(8) 


x= 3 


(6) 



We multiply (2) by 6 and (1) by 8, and obtain (3) and. (4); 
subtracting (4) from (3) we have (5), which reduced gives (6), or 
X = 3. Substituting this value of x in (1), we have (7), which re- 
duced gives (8), or y = 4. 

2. There is a fraction such that if 2 is added to the 
numerator the fraction will be equal to ^ ; but if 3 is 
added to the denominator the fraction will be equal to ^. 
What is the fraction ? 

SOLUTION. 

Let - = the fraction. 

y 

Then, by the conditions, 

'-±1 = \ (1) and^,=i (2) 

Zx = y-\-^ (3) 

2x + 4=y (4) 

X — 4 = 3 (5) 

x=1 (6) 

14 + 4 = 18=y a) ^ = f, (8) 

Clearing (1) and (2) of fractions, we obtain (3) and (4) ; sub- 
tracting (4) from (3), we obtain (5), which reduced gives (G), or 
a; = 7. Substituting this value of x in (4), we have (7), ot y= 18. 

Hence, - - ~ 



114 ELEMENTARY ALGEBRA. 

3. There are two numbers whose sum is 28, and one 
fourth of the first is 3 less than one fourth of the second. 
What are the numbers ? Ans. 8 and 20. 

4. The ages of two persons, A and B, are such that 5 
years ago B's age was three times A's ; but 15 years hence 
B's age will be double A's. What is the age of each ? 

Ans. A's, 25; B's, 65. 

6. There are two numbers such that one third of the 
first added to one eighth of the second gives 39 ; and 
four times the first minus five times the second is zero. 
What are the numbers ? 

6. Find a fraction such that if 6 is added to the nu- 
'nerator its value will be ^, but if 3 be added to the de- 
nominator its value will be -^ ? Ans. /j. 

t. What are the two numbers whose difference is to 
their sum as 1:2, and whose sum is to their product 
as 4 : 3 ? 

SOLUTION. 

Let X = the greater and y = the less. 

Thenx — i/:x + y=l:2 (1) x-]-y : xy = 4.: S (2) 

2x — 2y = x + y (3) Sx + Sy = 4:xy (4) 

x = Sy (5) 9y + Sy=l2f (6) 

x = 3 (1) l=:y (8) 

Having written (1) and (2) in accordance with the statement in 
the problem, we form from them (3) and (4) by Art. 106. Re- 
ducing (3), we obtain (5) ; substituting this value of x in (4), we 
have (6), which, though an equation of the second degree, can be 
at once reduced to an equation of the first degree by dividing each 
term by y ; performing this division and reducing, we obtain (8) or 
y = 1 ; substituting this value of y in (5) we obtain (7), or ar = 3. 



EQUATIONS OF THE FIRST DEGREE. 115 

8. What are the two numbers whose difference is to 
their sum as 3 : 20, and three times the greater minus twice 
the less is 35 ? 

9. There is a niimb^r consisting of two figures, which 
is seven times the sum of its figures ; and if 36 is sub- 
tracted from it, the order of the figures will be inverted. 
What is the number ? Ans. 81. 

10. There is a number consisting of two figures, the 
first of which is the greater ; and if it is divided by the 
sum of its figures, the quotient is 6 ; and if the order of 
the figures is inverted, and the resulting number divided 
by the difference of its figures plus 4, the quotient will 
be 9. What is the number? Ans. 54. 

11. As John and James were talking of their money, 
John said to James, " Give me 15 cents, and I shall have 
four times as much as you will have left." James skid 
to John, "Give me 7^ cents, and I shall have as much 
as you will have left/' How many cents did each 
have ? Ans. John, 45 cents ; James, 30 cents. 

12. The height of two trees is such that one third of 
the height of the shorter added to three times that of 
the taller is 360 feet ; and if three times the height of 
the shorter is subtracted from four times that of the taller, 
and the remainder divided by 10, the quotient is 17. Re- 
quired the height of each tree. 

Ans. 90 and 110 feet. 

13. A farmer who had $41 in his purse gave to each 
man among his laborers $2.50, to each boy SI, and had 
$15 left. If he had given each man S4 and then each 
boy $3 as long as his money lasted, 3 boys would have 
received nothing. How many men and how many boys 
did he hire ? 



116 ELEMENTAKY ALGEBRA. 

14. A man worked 10 days and his son 6, and they 
received $ 31 ; at another time he worked 9 days and 
Ms son 1, and they received $29.50. What were the 
wages of each ? 

m 

15. A said to B, "Lend me one fourth of your money, 
and I can pay my debts." B replied, "Lend me $100 
less than one half of yours, and I can pay mine." Now 
A owed $1200 and B $1900. IIow much money did 
each have in his possession ? 

Ans. A, $800 ; B, $1600. 

16. If a is added to the difference of two quantities, 

the sum is b ; and if the greater is divided by the less, 

the quotient will be c. What are the quantities ? 

. he — ac , h — a 

Ans. ■ - and -• 

c — 1 c — 1 

lY. A man owns two pieces of land. Tliree fourths 
of the area of the first piece minus two fifths of the area 
Df the second is 12 acres ; and five eighths of the area 
of the first is equal to four ninths of the area of the 
second. How many acres are there in each ? 

Ans. 1st, 64 acres ; 2d, 90 acres. 

18. A and B begin business with different sums of 
money; A gains the first year $350, and B loses $500, 
and then A's stock is to B's as 9 : 10. If A had lost 
$500 and B gained $350, A's stock would have been to 
B's as 1:3. With what sum did each begin ? 

Ans. A, $1450; B, $2500. 

19. If a certain rectangular field were 4 feet longer 
and 6 feet broader, it would contain 168 square feet more ; 
but if it were 6 feet longer and 4 feet broader, it would 
contain 160 square feet more. Required its length and 
breadth. 



EQUATIONS OF THE FIRST DEGREE. 117 

20. A market-man bought eggs, some at 3 for Y cents 
and some at 2 for 5 cents, and paid for the whole $2.62 ; 
he afterward sold them at 36 cents a dozen, clearing 
$0.62. How many of each kind did he buy? 

21. A and B can perform a piece of work together in 
12 days. They work together 7 days, and then A fin- 
ishes the woRk alone in 15 days. How long would it 
take each to do the work ? Ans. A 36 and B 18 days. 

22. "I was ten times as old as you 12 years ago,'' 
said a father to his son ; " but 3 years hence I shall be 
only two and one half times as old as you.'' What 
was the age of each ? 

23. If 3 is added to the numerator of a certain frac- 
tion, its value will be f ; and if 4 is subtracted from the 
denominator, its value will be j-. What is the fraction? 

24. A farmer sold to one man Y bushels of oats and 5 
bushels of corn for $12.76, and to another, at the same 
rate, 5 bushels of oats and 7 bushels of corn for $13.40. 
What was the price of each ? 

25. Find two quantities such that one third of the first 

minus one half the second shall equal one sixth of a ; 

and one fourth of the first plus one fifth of the second 

shall equal one half of a. . 34 a , 15 a 

Ans. — and -23^. 

26. A person had a certain quantity of wine in two 
casks. In order to obtain an equal quantity in each, he 
poured from the first into the second as much .as the 
second already contained; then he poured from the sec- 
ond into the first as much as the first then contained ; 
and, lastly, he poured from the first into the second as 
much as the second still contained ; and then he had 16 
gallons in each cask. How many gallons did each origi- 
nally contain? Ans. 1st, 22 ; 2d, 10 gallons. 



118 



ELEMENTARY ALGEBRA. 



SECTION XV. 

EQUATIONS 

OF THE FIRST DEGREE CONTAINING MORE THAN TWO 
UNKNOWN QUANTITIES. 

117. The methods of elimination given for solving equa- 
tions containing two unknown quantities apply equally 
well to those containing more than two unknown quantities. 



( ^+ y— ^= 4 

1. Given -l^x -\- ^y -{- Az 
izx — 2y\-bz 



= A. 

= n k to 
= 5) 



find X, y, and z. 



OPERATION. 






a: + y — 2 = 4 (1) 2x + 3y + 4.z=l7 (2) 


3x 


— 2r/+ 5z= 5 (3) 


2x + 2y — 2z= 8 (4) 


3x 


+ 3y— 32; =12 (5,^ 


y + 62= 9 (6) 




by— 82= 7 (7{ 
5y + 302=45 (8j 


x + 3 — 1=4 (13) y + 6 = 9 (11) 




382 = 38 (9) 


x = 2 ' (14) y = S (12) 




z= 1(10) 



Multiplying equation (1) by 2 gives equation (4), which we sub- 
tract from (2), and obtain (6) ; multiplying (1) by 3 gives (5), and 
subtracting (5) from (3) gives (7). We have now obtained two 
equations, (6) and (7), containing but two unknown quantities. Mul- 
tiplying (6) by 5, we obtain (8), and subtracting (7) from (8), we 
obtain (9), which reduced gives 2=1. Substituting this value of 
z in (6), and reducing, we obtain y == 3. Substituting these values 
of y and z in (1), and reducing, we obtain x = 2. 



2. Given - 



X + y =26' 
y +z=29 
z -|- t^> = 66 
w -\-u =Sl 

M + « = 4:6 



■ , to find u, w, X, y, and z. 



EQUATIONS OF THE FIRST DEGREE. 119 

OPERATION. 

z + y«26 (1) y + « = 29 (2) *-Hw-56 (8) ir + « =« 81 (4) « + x»-46 (5| 

y + z = 28 «— z— 3 to+z=53 «— z = 28 



X— z- 8 (6) i»+z=53 (7) « — z=28 (8) 2z = 18 (9) 

y = 17 (11) z = 12 (12) ir = 44 (13) u = 37 (14) z= 9(10) 

Here we subtract (1) fipom (2), and obtain (6) ; then (6) from 
(3), and obtain (7) ; then (7) from (4), and obtain (8) ; then (8) 
from (5), and obtain (9), which reduced gives (10), or x = 9. Sub- 
stituting this value of x in (1), (6), (7), and (8), and reducing, we 
obtain (11), (12), (13), and (14), or y = 17, z = 12, w = 44, and 
w = 37. 

Hence, for solving equations containing any number 
of unknown quantities, 



RULE. 

Fr(mi the given equations deduce equations one less in 
number, containing one less unknown quantify; and con- 
tinue thus to eliminate one unknown quantity after an- 
other , until one equation is obtained containing but one 
unknown quantity. Reduce this last equation so as to find 
the value of this unknown quantity ; then substitute this value 
in an equation containing this and but one other unknown 
quantity, and reducing the resulting equation, find the value 
of this second unknown quantity ; substitute again these values 
in an equation containing no more than these two and one 
otJier unknown quantity, and reduce as before ; and so con- 
tinue, till the value of each unknown quantity is found. 

Note. — The process can often be very much abridged by the 
exercise of judgment in selecting the quantity to be eliminated, the 
equations from which the other equations are to be deduced, the 
method of elimination which shall be used, and the simplest equa- 
tions in which to substitute the values of the quantities which have 
been found. 



120 



ELEMENTARY ALGEBRA. 



Find the values of the unknown quantities in the fol- 
lowing equations : — 



3. Given 



X -{- f/ -\- z -{- w = 16" 
x-{-i/-\-w-\-u= 14 

X-\-1/-{-Z-\- It z=l[>^ 



Note. — If these equations are added together and the sum di-^ 
vided by 4, we shall have x -\- y -{- z -{- w -\- u = 20 ; and if 
from this the given equations are successively subtracted, the values 
of the unknown quantities become known. r ^ g 

U = 3. 
Ans.<i z =z. Q, 



6. 



4. Given 




x=2. 



Ans. -J y = 4:. 

(z=:6. 



6. Given - 



^2x + 33/ + 4^ = 6T-] 
4 



2 "> 2 
2y + ^ 



'ZbJ 



Ans 



rx= 1. 
iz = n. 



6. Given 



a;— y — 2;=: 1 

X-{-2l/— 10z=: 1 

2x — 4y-f- 3s= 1 



rx =z 5. 
Ans. -|y r= 3. 

U = i. 



T. Given 



\^+y+\^=22 



1 ,1,1 
i^ + iy + 2^ 

1 , 1 . , 1 



24 



10 



Ans. 



E 



20. 
12. 
32. 



EQUATIONS OF THE FIBST DEGREE. 



121 



8. Given ^ 



1,1 5 

x'^y— 6 

y^2 12 

1 , 1 ___ 3 



Note. — The best method for this example is that used in Ex- 
ample 3, without clearing of fractions. 



9. Given 



( ^+ y+ ^= 6j fxz= 

-] 2a: + 3y + 4^ = 20 [■ . Ans. U = 



xz= 1. 
2. 
3. 



10. Given 



11. Given 



rx + iy = 3U 
]y + ^z = ll[ 
iz +ix = 29) 



x + y = a 



11 



rx-^y = a^ /a? = ^ (a — ft -j- c) 

■\y + ^ = f>y- Ans. }y = i{a + b—c) 
(a: + z = c) iz=^{b-\-c — a) 

M 

Ans. < f/z=- 



abx-\-abyz=:a-\- b 



raox-^a()y=za-\- b ^ 

Given <acx-{'acz=za-\-c> 

\bcy-\-bcz:=b +c) 



13. Given 



I 



x+ 21 =^y+28 
9x = 2y 



\ 



122 ELEMENTARY ALGEBBA. 

PROBLEMS 

PRODUCING EQUATIONS OF THE FIRST DEGREE CON- 
TAINING MORE THAN TWO UNKNOWN QUANTITIES. 

118. 1. A merchant has three kinds of flour. He can 
sell 1 bbl. of the first, 2 of the second, and 3 of the third 
for $85; 2 of the first, 1 of the second, and ^ bbl. of 
the third for $45.50 ; and 1 of each kind for $41. What 
is the price per bbl. of each ? 

Ans. 1st, $ 12 ; 2d, $ 14 ; 3d, $ 15. 

2. Three boj'^s. A, B, and C, divided a sum of money 
among themselves in such a manner that A and B re- 
ceived 18 cents, B and C 14 cents, and A and C 16. How 
much did each receive? Ans. A, 10 ; B, 8 ; C, 6 cents. 

3. As three persons, A, B, and C, were talking of their 
ages, it was found that the sum of one half of A's age, one 
third of B's, and one fourth of C's was 33 ; that the sum 
of A's and B's was 13 more than C's age ; while the sum 
of B's and C's was 3 less than twice A'a age. What 
was the age of each? Ans. A's, 32 ; B's, 21 ; C's, 40. 

4. As three drovers were talking of their sheep, says 
A to B, "If you will give me 10 of yours, and C one 
fourth of his, I shall have 6 more than C now has." 
Says B to C, *'If you will give me 25 of yours, and A 
one fifth of his, I shall have 8 more than both of you 
will have left." Saj^s C to A and B, "If one of you 
will give me 10, and the other 9, I shall have just as 
many as both of you will have left." How many did 
each have? 

5. Divide 32 into four such parts that if the first part 
is increased by 3, the second diminished by 3, the third 
multiplied by 3, and the fourth divided by 3, the sum, 
difierence, product, and quotient shall all be equal. 

Ans. 3, 9, 2, and 18. 



EQUATIONS OF THE FIRST DEGREE. 123 

6. If A and B can perform a piece of work together 
in 8|2j. days, B and C in 9^r^ days, and A and C in 8J 
days, in how many days can each do it alone? 

Ans. A in 15, B in 18, and C in 21 days. 

7. Find three numbers such that one half of the fijst, 
one third of the second, and one fourth of the third shall 
together be 56 ; one third of the first, one fourth of the 
second, and one fifth of the third, 43 ; one fourth of the 
first, one fifth of the second, and one sixth of the third, 35. 

8. The sum of the three figures of a certain number is 
12 ; the sum of the last two figures is double the first ; 
and if 297 is added to the number, the order of its fig- 
ures will be inverted. What is the number? 

Ans. 417. 

9. A man sold his horse, carriage, and harness for 
$450. For the horse he received $25 less than five 
times what he received for the harness ; and one third 
of what he received for the horse was equal to what he 
received for the harness plus one seventh of what he 
received for the carriage. What did he receive for each? 

Ans. Horse, $225; carriage, $175; harness, $50. 

10. A man owned three horses, and a saddle which 
was worth $45. If the saddle is put on the first horse, 
the value of both will be $30 less than the value of the 
'second ; if the saddle is put on the second horse, the value 
of both will be $55 less than the value of the third ; and 
if the saddle is put on the third horse, the value of both 
will be equal to twice the value of the second minus $ 10 
more than one fifth of the value of the first. What is 
the value of each horse ? 

Ans. Ist, $100; 2d, $175; 3d, $275. 

11. The sum of the numerators of two fractions is 7, 
and the sum of their denominators 16 ; moreover the sum 
of the numerator and denominator of the first is equal 



124 ELEMENTARY ALGEBRA. 

to the denominator of the second ; and the denominator 
of the second, minus twice the numerator of the first, is 
equal to the numerator of the second. What are the 
fractions ? Ans. f and f . 

l/Q. A man bought a horse, a wagon, and a harness, 
for $ 180. The horse and harness cost three times as 
much as the wagon, and the wagon and harness one half 
as much as the horse. What was the cost of each ? 

13. A gentleman gives $600 to be divided among three 
classes in such a way that each one of the best class is 
to receive $10, and the remainder to be divided equally 
among those of the other two classes. If the first class 
proves to be the best, each one of the other two classes 
will receive $5 ; if the second class proves to be the best. 
each one of the other two classes will receive $4f ; but 
if the third class proves to be the best, each one of the 
other two classes will receive $2, What is the number 
in each class? 

14. A cistern has 3 pipes opening into it. If the first 
should be closed, the cistern would be filled in 20 min- 
utes ; if the second, in 25 minutes ; and if the third, in 
80 minutes. How long would it take each pipe alone 
to fill the cistern, and how long would it take the three 
together ? 

Ans. 1st, 85f minutes ; 2d, 46-^7j- minutes ; 3d, 35^^^ 
minutes. The three together, IQ-^r minutes. 

15. Three men. A, B, and C, had together $24. Now 
if A gives to B and C as much as they already have 
and then B gives to A and C as much as they have after 
the first distribution, and again C gives to A and B as 
much as they have after the second distribution, they will 
all have the same sum. How much did each have at 
first? Ans. A, $13 ; B, $1, and C, $4. 



EQUATIONS OF THE FIRST DEGREE. 125 

SECTION XYI. 

POWERS AND ROOTS. 

119. A Power of any quantity is the product obtained 
by taking that quantity any number of times as a factor; 
and the exponent shows how many times the quantity is 
taken (Art. 24). Thus, 

a=ia^ is the first power of a ; 
a a=Ma^ " . second power, or square, of a ; 
a a a:=: a^ " third power, or cube, of a ; 
a a a az:^ a^ " fourth power of a ; 
and so on. 

Though not strictly within the definition, the quantity 
itself is called the first power, and the first root of itself. 

120. In order to explain the use of negative indices, 
we form, by the rules of division, the following series: — 

' 8 4 3 2 1^1 111 

^' «' «' «' «' ^' a' ^' ^' a<' d'^ 
a\ a\ a», a^ a\ a^ a-\ a"*, a"*, a'^ a-\ 

We form the first series as follows: a* divided by a gives a*; 
a* by a, gives a*; a' by a, gives a'; a' by a, gives a; a by a, gives 

1 ; 1 by a, gives -; - by a, gives ,; , by a, gives -j, and so on. 

The second series is formed in the same way from a' to a; but if 
wc follow the same rule of division from a toward the right as from 
rt* to a, viz. subtracting the index of the divvsor from that of the div- 
l^nd^ a divided by a, gives cfi\ cfi by a, gives a-^ ; read a, with the 
negative index one; a~* by a, gives o""; a~* by a, gives a~*\ and 
Bo on. 

From this we learn, 

Ist. That the power of every quantity is 1 ; 

2d. That cr^, a"*, ar^, &c., are only different ways of 

•^ 111^ 

wrxtinq -, -,, — ,» (Jcc. 

^ a or or 



126 ELEMENTARY ALGEBRA. 

Any two quantities at equal distances on opposite sides 
of a°, or 1, are reciprocals of each other. 

121. The rules given for the multiplication and divis- 
ion of powers of the same quantity (Arts. 50 and 54) 
apply equally well whether the exponents are positive 
or negative. For 

1 a' 
a^ X ar^ = a^ X -, = ~, = ci^ 



a« -T- a-^ = a^ -f- - = a^ X ci' = c^ 



2_^ 



a' 



a' a" 

The following examples in multiplication are to be done 
according to the rules for the multiplication of powers of 
the same quantity by each other, given in Art. 50 ; and 
those in division, by the rule for the division of powers 
of the same quantity by each other, given in Art. 5JL. 

1. Multiply x' by x'^. Ans. x^. 

2. Multiply a^ by a'^. 

3. Multiply x^ by x'^. Ans. a;^ or 1. 

4. Multiply y-'^ by y*. 

5. Multiply a-^x^y^ by a~^x~^y'^. 

Ans. ar^x^y^, or ar^y''. 

6. Multiply Ix-^y-^z by 3xV^*- 

1. Multiply llx'^fz-^ by A^x-^y-^^^, 

8. Multiply ^ll|-^' by 5 a^ b-"- c\ 

9. Divide x^ by x~^. Ans. x^^. 

10. Divide x^ by x''^. 

11. Divide a:~" by x~^. Ans. x-*. 



EQUATIONS OP THE FIBST DEGREE. 127 

12. Divide y-^ by f. 

13. Divide y' by y*. Ans. y'^. 

14. Divide a~*6c^ by a^h~* c'^. Ans. a~'^U*c*. 

15. Divide 16x'y-*2 by 4a;-y-*2:«. 

16. Divide 4ar»y-»z by 2aar-2y-«2*. 

IT. Divide *la^hx-^y^ by 10 a i*^ a:* y-«2«. 

18. Divide 144<7«6c-^x*y-'2 by 16 «''6-^c-='ar^/. 

122t It follows from the preceding article that a factor 
may he transferred from the numerator of a fraction to its 
denominator, or vice versa, provided the sign of the expo- 
nent of the factor is changed from -\- to — , or — to +. For 

^ = a« X ^4 = a« X a:-* = a'x-* 

a a 1 v> _^ ^ 

-,= ^=a^-, = aX^ = a2/ 

y ""y ^ ""y * ar'^y * "^ "~x-^y 
— = 1 X - = — 

X (/ ^ X dx 

1. Transfer the denominator of ^-^ — r to the numerator. 

bc'y-'^ 

Ans. cU Ir^ c'"^ x^ y . 

2. Transfer the numerator of -j- , - , to the denommator. 

3. Transfer the denominator of /, . to the numerator. 

a <r 

4. Transfer the numerator of ° . ^ ^ to the denominator. 



128 ELEMENTARY ALGEBRA. 

6. Free from negative exponents ,_^ —4. 

Ans. ■= — I — ^- 
lac*xy 



6. Free from negative exponents 
1. Free from negative exponents 






8. Free from negative exponents . ::_a /"^ x • 






3.— n yn 



9. Free from negative exponents -zitj:- 

INVOLUTION. 

123. Involution is the process of raising a quantity to 
a power. 

124. A quantity is involved by taking it as a factor as 
many times as there are units in the index of the re- 
quired power. 

125. According to Art. 48, 

(+«) X (+a) X (+a) = (+a^) X {+a)= + a\ 
and so on ; 

and ( — a ) X ( — a)= + fl^^ 

(-a) X (-«) X (-«) = (+a') X {-a) = -a\ 
(—a) X (— «) X i—a) X (— «) = (—a') X {—a)= + a\ 
and so on. 

Hence, for the signs we have the following 

RULE. 
Of a positive quantity all the powers are positive. 
Of a negative quantity the even powers are positive, and 
the odd povjers negative. 



INVOLUTION. 129 

INVOLUTION OF MONOMIALS. 

126. To raise a monomial to any required power. 

1. Find the third power of 2 a* 6. 

OPERATION. 

(2 a» J)» = 2 an X 2 a« i X 2 a« 6 (1) 

= 2. 2. 2. a'^a^'a^bbb (2) 

= 8a»6» (3) 

According to Art 124, to raise 2 0*6 to the third power we take 
it as a factor three times (1) ; and as it makes no difference in the 
product in what order the factors are taken, we arrange them as 
in (2) ; performing the multiplication (Art. 60) expressed in (2), 
we have (3). Hence, 

RULE. 
Multiply the exponent of each letter by the index of the 
required power, and prefix the required power of the nu- 
merical coefficient, remembering that the odd powers of a 
negative quantity are negative, while all other powers are 
positive. 

Note. — It follows that the power of the product is equal to the 
product of the powers. 

2. Find the square of 2 x. Ans. 4 x*. 

3. Find the cube of 3 x^. Ans. 21 a:«. 

4. Find the fourth power of a* 6*. Ans. a" ft*. 
6. Find the third power of 4 a'' a:. Ans. 64 a* a:*. 

6. Find the square of 2 x'^. Ans. 4 x'^, 

7. Find the cube of 3ar-*y^ Ans. 27a:-»y. 

8. Find the mth power of a b. Ans. a*" ft'\ 

9. Find the third power of — 3 a'* b. Ans. — 27 a' ft^ 



130 ELEMENTARY ALGEBRA. 

10. Expand (—2a^xy. Ans. 16a^^x\ 

11. Expand (3 aH")"*. Ans. S^a^^If'\ 

12. Expand (2x'yy. 

13. Expand {—la'^ot^y, 

14. Expand (--3a;»y)8. 

15. Expand (— a-y. Ans. — a-\ 

16. Expand (x-^y^. 

11. Expand (— 4a;-8 3/)2. Ans. i^. 

18. Expand (3a'»a;2)4. 

19. Expand (— 2 a;-^ 3/-«)». 

20. Expand (— 3a;-2» y'»)6. 

21. Expand (—9 a-2i-'»a;V)'- 

INVOLUTION OF FRACTIONS. 

127t To involve a fraction. 

a* 
1. Find the cube of ^r-r' 
2 



V2&7 ""26 



OPERATION. According to Art. 124, 

sy ^^ ^ ^ ^ *^ fi"^^ *^® ^^^ of ^'^y 

^ 2^ '^ 26 86» quantity we must take it 
three times as a factor: 

taking — =- three times as a factor, and performing the multiplica- 

a' 
tion by Art. 95, we have 77^5. Hence, 



BULE. 
Involve both numerator and denominator to the required 
power. 



INVOLUTION. 131 



2. Find the square of r— 3- 






3. Find the cube of — ir-r-,- Ans. — 



4. Find the fourth power of 



26c« 



3xr" 



6. Find the fifth power of — ^^'. 

2 a z~* 

6. Find the third power of 



7. Find the mth power of 






2 a'ar** 

8. Find the fourth power of , _, ^ « 

3 a* 6-* c-' 

9. Find the third power of — h'^^fr^' 

10. Find the fourth power of — -—_,-:_,-■ 

■^ m ■ n^ p 

11. Find the fifth power of — ~i'lr*'<*' 



INVOLUTION OF BINOMIALS. 

128. A BINOMIAL can be raised to any power by suc- 
cessive multiplications. But when a high power is re- 
quired, the operation is long and tedious. The Binomial 
Theorem, first developed by Sir Isaac Newton, enables us 
to expand a binomial to any power by a short and speedy 
process. 

129t In order to investigate the law which governs the 
expansion of a binomial we will expand a-\-h and a — h 
to the fifth power by multiplication. 



132 ELEMENTARY ALGEBRA. 

a -\-h 

a +h • 

a^-i- ab 

ah 4-y 

a^ -\- 2 a b -\- h^ 2d power. 

a +b 

an-\- ^ab^ -\-b^ 
a^j^Zan+ 3 a 5^ + 6« . . . .3d power. 
a -\-b 

an 4- 3 g^ ^2 _[_ 3 a 53 _[_ ^4 
a*-(-4an-|- 6a2^2_|_ 4^j3 _|_54 ^ ^ 4th power. 

g -f& 

a«_|-4:an+ 6an2+ 4a2 6«4- a6* 

a^-l- 4a«62 4- 6g2^«-j-4aZ^^ + ^« 
a5_j_6an 4- 10^6^+ l^ aH^ -\- b a ¥ -\- b^ 6th power. 

a — h • 

a — h 
o^ ■— a 6 

— g 6 + 6^ 

g2 — 2 a 6 + 62 . . , , . .2d power. 
a — h 

— g26-f- 2 g62 — 58 
aS _ 3 ^2 5 _j_ 3 gj 52 _ j8 

a — b 

ai^San+ 3g262_ ^Ti^ 

— g^ft-i- 3g2&2_ 3a6 « + &^ 

a4_4aH-f ^an^— 40^^^ - . 4th power. 

a — b 

a5 — 4an+ 6gH2_ 4,aH^+ a¥ 

•— g^-f 4an2_ 6a2&«-f 4a&^-~-&s 
o^ — 5 a^ 4- 10 a" ^' -- 10 a2 6^ + 5 a 6^ — 6^ 6th powen 



3d power. 



INVOLUTION. 13B 

By examining the diflferent powers of a -\- h and a — b 
in these Examples, we shall find the following invariable 
laws governing the expansion : — 

1st. The leading quantity (i. e. the first quantity of th^ 
binomial) begins in the first term of the power with an ex- 
ponent equal to the index of the power, and its exponent 
decreases regularly by one in each successive term till it dis- 
appears ; the following quantity (i. e. the second quantity 
of the binomial) begins in the second term of the power ivitJ. 
the exponent one, and. its exponent increases regularly hj 
one till in the last term it becomes the same as the index of 
the power. 

Thus, in the fifth power the 

Exponents of a are 5, 4, 3, 2, 1. 
Exponents of 6 are * 1, 2, 3, 4, 6. 

It will be noticed that the sum of the exponents of the 
letters in any term is equal to the index of the power. 

2d. The coefficient of the first term is one ; of the second, 
the same as the index of the power ; and universally, the co- 
efficient of any term, multiplied by the exponent of the lead- 
ing quantity, and this product, divided by the exponent of 
Die following quantity increased by one, will give the co- 
efficient of the succeeding term. 

Thus, in the fifth power, 5, the coefficient of the second 
term, multiplied by 4, a's exponent, and divided by 
1 plus 1, 6's exponent plus 1, =: — ^ = 10, the coeffi- 

cient of the third term. 

The coefficients are repeated in the inverse order after 
passing the middle term or terms, so that more than half 
of the coefficients can be written without calculation. The 
number of terms is always one more than the index of 



134 ELEMENTARY ALGEBRA. 

the power ; i. e. the second power has three terms ; the 
third power, four terras ; and so on. When the number 
of terms is even, i. e. when the index of the power is 
odd, the two central terms have the same coefficient. 

3d. When both terms of the binomial are positive, all the 
terms of the power are positive; but when the second term 
is negative, those terms which contain odd powers of the 
following quantity are negative, and all the others positive; 
or every alternate term, beginning with the second, is negor 
tive, and the others positive. 

1. Expand {x -{- yy. 

OPERATION. 

According to the law, the first term will be 
and the second term 



The coefficient of the third term will be 
and the third term 





A 




+ 8x^y. 


4 




^X 7 




2. ' 






+ 28^3^. 


2 




28 X^ 




X ' 






■\-^^^f- 


14 




^X5 





The coefficient of the fourth term will be 
and the fourth term 



The coefficient of the fifth term will be 

and the fifth term 70 a;* 2^*. 

Having found the preceding coefficients and the coefficient of thi 
middle term, we can write the others at once. Hence, 

(X + y)8 = z8 + 8x7y + 28x«y2 ^ 56z5y3 ^. 70x*y4 + 56x3y5 ^ 28x2y6 + ^xyi + yS. 

2. Expand {a — hy. 

Ans. a«— 6a5&+15a*62~20a353+15a2i*— 6a65+5«. 

3. Expand (m + '*)^. 

4. Expand {b — yy. 
6. Expand (a ■— a:)^^ 



INVOLUTION. 185 

6. Expand {b + cy\ 
T. Expand (x + 1)». 

Note. — Since all the powers of 1 are 1, 1 is not written when 
it appears as a factor; but its exponent must be used in obtaining 
the cocrticients. 

Ans. X* 4- 5 X* + 10 ar' + 10 x2 + 5 a: 4- 1. 

8. Expand (I — y)^ 

Ans. l — 6y + 15y2 — 203/^ + 153/* — 6/ + /. 

9. Expand (a -- l)\ 

130. When the terras of the binomial have coefficients 
or exponents other than 1, the theorem can be made to 
apply by treating each term as a single literal quantity. 
In the expansion, each factor should be enclosed in a 
parenthesis, and after the expansion of the binomial by 
the binomial theorem, the work should be completed by 
the expansion of the enclosed factors, according to the 
rule for the expansion of monomials. 

1. Expand (2x — fy. 

OPERATION. 

(2 xy - 4 (2 x)» (y«) + 6 (2 x)' (/)» - 4 (2 x) (f)' + (f)* 
Expanding each factor as indicated, we have 

16 ar* — 32 a:«y« + 24 ar^^ — 8 x/ + y« 

2. Expand (3x« — 2y)^ 

(8x«)s - 6(8x«)4 (2y) + 10(8z9)» (2yy> - 10(8a»r« (2y)» + 6(3z«) (2y)4 — (2y)5. 
Ans. 243x" — 810z»y-j-1080z*y>— 720x*y»-}-240x*y*— 32/. 

Note. — Any letters, as a and b, might be substituted for 3 a:" 
and 2 y, and the expansion of (a — by written out, and then the 
Talues of a and b substituted. 

3. Expand (a' — Sby. 

Ans. a'—UaH + 54ca*b^—lOSaH^ + Slb*. 

4. Expand {x^^y^y. 



136 ELEMENTARY ALGEBRA. 

5. Expand (2 a + iy. 

Ans. 8 a3+ 84 a'' + 294 a + 343. 

6. Expand (2ac — xy. 

Ans. 16a*c^ — 32a3c3a; + 24a2c2a:2 — 8aca;«+a:4. 

7. Expand (a^x--2yy. 



8. Expand (^ + xY. 

9. Expand (I- ly. 



^^«-il + I + ¥ + 2^« + x*. 



Ans ^__i^ f i^_5a^ 5aa;* a:* 
■ 32 48 "f" 36 54 »" "162 243 



10. Expand — 1 V. 

11. Expand (^^^_L.)'. 



27 "^ ^^ 8 64 



12. Expand (i + iy. 

13. Expand /ac— iV" 

14. Expand /^a: + -Y, 

15. Expand (l — -V. 

16. Expand /'2 a^ — IV. 
n. Expand (l+lY. 
18. Expand (i-iy. 



EVOLUTION. 137 

131* The Binomial Theorem can be applied to the ex- 
pansion of a polynomial. Thus, in a -\- b — c, a -\- b 
can be treated as a single term, and the quantity can be 
written (a -J- i) — c. In like manner, a -\- b -\- x — y can 
be written (a -]- b) -{- (x — y). In such cases it is easier 
to substitute a single letter for the enclosed terms, and 
after the expansion to substitute the proper values. 

1 . Expand (a + ^ — <?)*. 

OPERATION. 

Put a -\- b = X 

(x — c)» = aH» — Sar'c + Sxc^ — c» 
Substituting for x, its value, a -f- 6 ; 

(a + 6 — c)' = o8 + 8o«6 + 3a6a + &3 — 3a5c — 6a6 c — 36« c + 3ac« + 86c« — c» 

2. Expand (2a — b — c — dy, 

Note. — For 2a — b — c — d write (2 a — 6) — (c -f d). 

Ans. 4a«— 4a6-|-ft* — iac—4ad-\-2bc-\-2bd-\-c'-\-2cd-\-cP. 

3. Expand (3a:— ^y — a + by. 

4. Expand (i x — a + by. 

EVOLUTION. 

132. Evolution is the process of extracting a root of 
a quantity. It is the reverse of Involution. 

133* A ROOT of any quantity is a quantity' which taken 
as a factor a given number of times will produce the 
given quantity. 

The number of times the root is to be taken as a factor 
depends upon the name of the root. Thus, the second 
or square root of a quantity is a quantity which taken 
twice as a factor will produce the given quantity ; the 
third or cube root is a quantity which taken three times 
as a factor will produce the given quantity ; and so on. 



138 ELEMENTARY ALGEBRA. 

A Root is indicated by the radical sign m^ ^ or by a 
fractional exponent. Thus, 

\/x, or x^ indicates the square root of x. 



3/— i 

s/ X, or x^ 


tt 


(t 


cube 


y/x, or x'« 


tt 


ft 


mth 



ft H {I 



it it It 



134. A root and a power may be indicated at the same 

time. Thus, \^ x^, or x^ , indicates the cube root of the 
fourth power of x, or the fourth power of the cube root 
of X ] for a power of a 7'oot of a quantify is equal to the 

same 7V0t of the same power of the quantity. /^8^ or 8^ 
is the square of the cube root of 8, or the cube root of 
the square of 8, i. e. 4. 

135. A perfect power is a quantity whose root can be 
found. A perfect square is one whose square root can 
be found ; a perfect cube is one whose cube root can be 
found ; and so on. 

136. Since Evolution is the reverse of Involution, the 
rules for Evolution are derived at once from those of 
Involution. And therefore, as according to Art. 125 an 
odd power of any quantity has the same sign as the 
quantity itself, and an even power is always positive, 
we have for the signs in evolution the following 

RULE. 

An odd root of a quantity has the same sign as the quan- 
tity itself 

An even root of a positive quantity is either positive or 
negative. 

An even root of a negative quantity is impossible, or im- 
aginary. 



EVOLUTION. 139 

SQUARE ROOT OF NUMBERS. 

137. The Square Root of a number is a number which, 
taken twice as a factor, will produce the given number. 

]38« The square of a number has twice as many fibres 
08 the rooty oi' one less than twice as many. Thus, 

Roots, 1, 10, 100, 1000. 

Squares, 1, 100, 10000, 1000000. 

The square of any number less than 10 must be less than 100; 
but any number less than 10 is expressed by one figure, and any 
number less than 100 by less than three figures; i. e. the square of 
a number consisting of one figure is a number of either one or two 
figures. The stjuare of any number between 10 and 100 must be 
between 100 and 10000; i. e. must contain more than two figures 
and less than five. And the square of any number between 100 
and 1000 must contain more than four figures and less than seven. 

Hence, to ascertain the number of figures in the square 
root of a given number. 

Beginning at units, point off the number into periods of 
two figures each ; there will be as many figures in (lie root 
(W there are periods, and for the incomplete period at the 
left, if any, one more. 

139t To extract the square root of a number. 
1. Find the square root of 6329. 

From the preceding explanation, it is evident that the square 
root of 5329 is a number of two figures, and that the tens figure 
of the root is the square root of the greatest perfect square in 53 ; 
i. e. ^Ad, or 7. Now, if we represent the tens of the root by a 
and the units by h, a -\- b will represent the root ; and the given 
number will be 

(a -f 5)' = a« -f 2 a ft -|- 6». 

Now a' = 70» = 4900 ; 

therefore, 2 a 6 -f 6» = 5329 — 4900 = 429. 

But . 2a6-f 6»=(2a-f 6)6; 



140 ELEMENTARY ALGEBRA. 

If therefore 429 is divided hy 2 a -\- b, it will give h the units of 
the root. But b is unknown, and is small compared with 2 a ; 
we can therefore use 2 a = 140 as a trial divisor. 429 -^ 140, 
or 42 -|- 14 = 3, a number that cannot be too small but may be 
too great, because we have divided by 2 a instead of 2 a -j- &. 
Then b = 3, and 2 a -f 6 = 140 + 3 = 143, the true divisor ; and 
(2 a + 5) & = 143 X 3 == 429 ; and therefore 3 is the unit figure of 
the root, and 73 is the required root. The work will appear as 
follows : — 

OPERATION. 

5 3 2 9 (7 3 
49 

1 4 3) 4 2 9 
429 

Hence, to extract the square root of a number, 

RULE. 
Separate the given number into periods of two figures each, 
by placing a dot over units, hundreds, dbc. 

Find the greatest square in the left-hand period, and place 
its root at the right. 

Subtract the square of this root figure from the left-hand 
period, and to the remainder annex the next period for a 
dividend. 

Double the root already found for a trial divisor, and, 
omitting the right-hand figure of the dividend, divide, and 
place the quotient as the next figure of the root, and also at 
the right of the trial divisor for the true divisor. 

Multiply the true divisor by this new root figure, subtract 
the product from tlie dividend, and to the remainder annex 
the next period, for a new dividend. 

Double the part of the root already found for a trial di- 
visor, and proceed as before, until all the periods have been 
employed. 



EVOLUTION. 141 

Note 1. — When a root figure is 0, annex also to the trial di- 
visor, and bring down the next period to complete the new dividend. 

Note 2. — If there is a remainder, after using all the periods in 
the given example, the operation may be continued at pleasure by- 
annexing successive periods of ciphers as decimals. 

Note 3. — In extracting the root of any number, integral or deci- 
mal, place the first point over unit's place ; and in extracting the 
square root, over every second figure from this. If the last period in 
the decimal periods is not full, annex 0. 

2. Find the square root of 46225. 

We suppose at first that a rep- 
OPKRATlON. j^gj^^ ^jjg hundreds of the root, 

46225 (2 15 and b the tens ; proceeding as in 

4 Ex. 1, we have 21 in the root 

Then letting a represent the 
hundreds' and tens together, i. e. 
21 tens, and b the units, we have 
4 2 5) 2 I 2 5 2 a, the 2d trial divisor, = 42 

2 12 5 tens ; and therefore 6 = 5; and 

2 a -j- 6 = 425 ; and 215 is the 
required root. 

8. Find the square root of 5013,4. 

OPERATION. 

5013.46(7 0.805+ 

49 

140.8)113.40 
112.64 



41)62 
41 



141.605) .T60000 
.7 8025 

4. Find the square root of 288369. Ans. 537. 

5. Find the square root of 42849. Ans. 207. 

6. Find the square root of 173.261. Ans. 13.16+. 

7. Find the square root of .9. Ans. .948+. 



142 ELEMENTARY ALGEBRA. 

8. Find the square root of 2. Ans. 1.4142-f-. 

9. Find the square root of 484. 

10. Find the square root of 48.4. 

11. Find the square root of .064. 

12. Find the square root of .00016. 

Note. — As a fraction is involved by involving both numerator and 
denominator (Art. 127), the square root of a fraction is the square root 
of the numerator divided hy the square root of the denominator. 

13. What is the square root of | ? Ans. §. 

14. What is the square root of ^| ? 

15. What is the square root of -^jj ? -^^ = Z^. Ans. f . 

Note. — If both terms of the fraction are not perfect squares, 
and cannot be made so, reduce the fraction to a decimal, and then 
find the square root of the decimal. A mixed number must be re- 
duced to an improper fraction, or the fractional part to a decimal, 
before its root can be found. 

16. What is the square root of | ? Ans. .53-|-. 
1 1 . What is the square root of 2-fT ^ 

18. What is the square root of -^^ ? 

19. What is the square root of Tf ? 

CUBE ROOT OF NUMBERS. 

140. The Cube Root of a number is a number which, 
taken three times as a factor, will produce the given 
number. 

141. The cube of a number consists of three times as 
many figures as the root, or of one or two less than three 
times as many. 

Roots, 1, 10, 100, 1000. 

Cubes, 1, 1000, 1000000, 1000000000. 

The cube of any number less than 10 must be less than 1000; 
but any number less than 10 is expressed by one figure, and any 



EVOLUTION. 143 

number leas than 1000 by less than four figures; i. e. the cube of 
a number consisting of one figure is a number of less than four 
figures. The cube of any number between 10 and 100 must be be- 
tween 1000 and 1000000; i. e. must contain more than three figures 
and less than seven. And in the same way -we see that the cube 
of any number between 100 and 1000 must contain more than six 
figures and less than ten. 

Hence, to ascertain the number of figures in the cube 
root of a given number, 

Becfinning at units, point off the number into periods of 
three figures each ; there tvill be as many figures in the root 
as there are periods, and for tJie incomplete period at the 
left, if any, one more. 

142* To extract the cube root of a number. 

1. Find the cube root of 42875. 

From the preceding explanation, it is evident that the cube root 
of 42873 is a number of two figures, and that the tens figure of 
the root is the cube root of the greatest perfect cube in 42 ; i. e. 
^ 27, oi*8. Now, if we represent the tens of the root by a and the 
units by 6, a -\- b will represent the root, and the given number 
will be 

(a -I- 6)» = a» + 3 a« & + 3 a &« + 6». 

Now a* = 30' = 27000; 

therefore, 3 a* 6 + 3 a 6« + 6* = 42875 — 27000 == 15875. 

But 3a>&-|-8aft«-f 6» = (3a*-f3a&-l-i»)6. 

If therefore 15875 is divided by 3 a' + 3 a 6 -f &» it will give 6, 
tlie units of the root. But 6, and therefore 3 a 6 -j- &*, a part of 
the divisor, is unknown, and we must use 3a^ = 2700 as a trial 
divisor. 15875 -^ 2700, or 158 -^ 27 = 5, a number that cannot 
be too small but may be too great, because we have divided by 3 a* 
instead of the true divisor, 3 a^ -{- 3 a b -\- l^. Then b = 5, and 
3a'-|- 3aft-f^= 2700 -f- 450 -f- 25 = 3175, the true divisor; 
and (3 a' 4- 3 a 6 -f- />») 6 = 31 75 X 5 = 15875, and therefore 5 is 
the unit's figure of the root, and 35 is the required root The work 
will appear as follows : — 



144 ELEMENTARY ALGEBRA. 

OPERATION. 

4 2 8 T 5 (35 Root. 

21 

Trial divisor, 3 a^ = 2 Y ' 

3ab=z 450 
b^=z 2 5 



True divisor, B a^ + 3ab + b^ = 3 1 75 



1 6 8 1 5 Dividend. 



158T5 



Hence, to extract the cube root of a number, 

RULE. 

Separate the number into periods of three figures each, by 
placing a dot over units, thousands, Sc. 

Find the greatest cube in the left-hand period, and place 
its root at the right. 

Subtract this cube from the left-hand period, and to the re- 
mainder annex the next period for a dividend. 

Square the root figure, annex two ciphers, and multiply 
this result by three for a trial divisor ; divide the dividend 
by the trial divisor, and place the quotient as the next figure 
of the root. 

Multiply this root figure by the part of the root previously 
obtained, annex one cipher and multiply this result by three ; 
add the last product and the square of the last root figure to 
the trial divisor, and the sum will be the true divisor. 

Multiply the true divisor by the last root figure, subtract 
the product from the dividend, and to the remainder annex 
the next period for a dividend. 

Find a new trial divisor, and proceed as before, until all 
the periods have been employed. 

Note 1. — The notes under the rule in square root (Art. 139) 
apply also to the extraction of the cube root, except that 00 must 
be annexed to the trial divisor when the root figure is 0, and after 
placing the first point over units the point must be placed over 
every third figure from this. 

Note 2. — As the trial divisor may be much less than the true 



EVOLUTION. 



145 



divisor, the quotient is frequently too great, and a less number 
must be placed in the root. 

2. Find the cube root of 1819144T. 



OPERATION. 



l8t Trial Divisor, 8a* =1200] 
Sab =- 360 
6« = 36 

l8t True Divisor, 3a'-|-3a& + 6» 

2d Trial Divisor, 



18191447 (2 63 

_8 

10191 1st Dividend. 



1596J 

8c^ =202800 
Sab = 2340 
6^= 9 



9576 



2d True Divisor, 3a«-f3a6-f-6»=205149 



61544 7 2d Div. 



615447 



We suppose at first that a represents the hundreds of the root 
and 6 the tens: proceeding as in Ex. 1, we have 26 in the root 
Then letting a represent the hundreds and tens together, i. e. 26 
tens, and b the units, we have 3 a', the 2d trial divisor, = 202800 ; 
and therefore 6 = 3; and 3 a^ -\- S ab -\- b*, the 2d true divisor, 
= 205149; and 263 is the required root. 

Note. — Though the 1st trial divisor is contained more than 8 
times in the dividend, yet the root figure is only 6. 

3. Find the cube root of 68116.47. 

OPERATION, 



6 8116.4 7 0(4 0.9 5+ 
64 



48 0.0 

10 8.0 

.8 1 



4 7 1 6.4 7 



49 8.8 1 

50 18.4 30 

6.13 50 

.0025 

50 24.56 7 5 



4417.9 2 9 



2 9 8.5410 00 



251.228375 
4 7.3126 25^ 



146 ELEMENTARY ALGEBRA. 

4. Find the cube root of 292420Y. Ans. 143. 

5. Find the cube root of 8120601. Ans. 201. 

6. Find the cube root of 36926037. 

7. Find the cube root of 67911.312. 

8. Find the cube root of 46417.8. 

9. Find the cube root of .8. Ans. .928+. 

10. Find the cube root of .17164. 

11. Find the cube root of .0064. 

12. Find the cube root of 25.0001Y. 

13. Find the cube root of 2.7. 

Note. — As a fraction is involved by Involving both numerator and 
denominator (Art. 127), the cube root of a fraction is the cube root 
of the numerator divided by the cube root of the denominator. 

14. What is the cube root 2 r ? Ans. f. 

15. What is the cube root of //^ ? 

16. What is the cube root of f||? |-f| == |f|. 

Ans. f. 

Note. — If both terms of the fraction are not perfect cubes, and 
cannot be made so, reduce the fraction to a decimal, and then find 
the cube root of the decimal. A mixed number must be reduced 
to an improper fraction, or the fractional part to a decimal, be- 
fore its root can be found. 

17. What is the cube root of ■^\? Ans. .899+. 

18. What is the cube root of ^j? 

19. What is the cube root of 3^ ? 

20. What is the cube root of 117?? 



EVOLUTION. 14T 

EVOLUTION OP MONOMIALS. 

1I3* As Evolution is the reverse of Involution, and 
since to involve a monomial (Art. 126) we multiply the 
exponent of each letter by the index of the required 
power, and prefix the required power of the numerical 
coefficient, 

Hence, to find the root of a monomial, 

RULE. 
Divide the exponent of each letter by the index of the re- 
quired root, and prefix the required root of the numerical 
cvefficient. 

Note 1. — The rule for the signs is given in Art. 136. As an 
even root of a positive quantity may be either positive or negative, 
we prefix to such a root the sign ± ; read, plus or minus. 

Note 2. — It follows from this rule that the root of the product 
of several factors is equal to the vroduct of the roots. Thus, 
V/"36 = v'l V^'O = 6. 

1. Find the cube root of So:*/. Ans. 2xy^. 

2. Find the square root of 4 x^. Ans. ±2x. . 

3. Find the third root of — 125 a«ar. 

Ans. — 5 a^x^. 

4. Find the fourth root of 81a"* b. 

Ans. ± 3 a-* b^. 

5. Find the fifth root of 32 a^^ b\ Ans. 2 a* bK 

6. Find the cube root of — 729 arV- 

Ans. — 9x1^. 

7. Find the fourth root of 256 xV- 

8. Find the cube root of — 512 a-H'. 

9. Find the fifth root of 243 a:* y^. 



148 ELEMENTARY ALGEBRA. 

Note. — As a fraction is involved by involving both numerator 
and denominator (Art. 127), a fraction must be evolved by evolving 
both numerator and denominator. 

4 q2 2 a 

10. Find the square root of r— r* Ans. ± ^r^,- 

Perform the operations indicated in the following ex- 
pressions : — 



11. \/—129aH^c\ 

12. (ida^x^f)^, 

13. 



/ 9 a? X* 
Y 36Z>V 



14. it^a'^x" 

15. {25Qa''x^'y^Y. 



,4^10^,16\-J 



16. /^Slan\ 



11. \^a'"62"*c"»". 



SQUARE ROOT OF POLYNOMIALS. 

144. In order to discover a method for extracting the 
square root of a polynomial, we will consider the rela- 
tion of a 4" ^ ^^ ^ts square, a^ ~\- 2 a b -{- h^. The first 
term of the square contains the square of the first term 
of the root ; therefore the square root of the first term of 
the square will be the first term of the root. The second 
term of the square contains twice the product of the two 
terms of the root ; therefore, if the second term of the 
square, 2 a b, is divided by twice the first term of the 
root, 2 a, we shall have the second term of the root b. 
Now, 2 a b -\- b^ = (2 a -\- b) b', therefore, if to the trial 
divisor 2 a we add b, when it has been found, and then 



EVOLUTION. 149 

multiply the corrected divisor by h, the product will be 
equal to the remaining terras of the power after a^ has 
been subtracted. 

The process will appear as follows : — 

OPERATION. Having written a, the square 

a^-\-2ab-\-lt^{a-\-b root of a", in the root, we sub- 

a^ tract its square (a') from the 

2a-|-i)2a64-^ g»ven polynomial, and have 

\ , \ VI 2ah 4- h" left. Dividing the 

' first term ol this remamder, 

2 a h, by 2 a, which is double the term of the root already found, 
we obtain ft, the second term of the root, which we add both to 
the root and to the divisor. If the product of this corrected divisor 
and the last term of the root is subtracted from 2 a ft -j- 6', nothing 
remains. 

145* Since a polynomial can always be written and 
involved like a binomial, as shown in Art. 131, we can 
apply the process explained in the preceding Article to 
finding the root, when this root consists of any number 
of terms. 

1. Findthe8quarerootofa2+2a5+ft=* — 2ac— 2ftc + (r». 

OPERATION. 

a^^2ab + P—2ac — 2bc + c^(a + b^c 



2a + b)2ab + lP 
2ab + if' 



2a + 2b — c)—2ac — 2bc + c* 
-^2ac — 2bc-\-(^ 



Proceeding as before, we find the first two terms of the root a-\-b. 
Considering a -\- h zr 2^ single quantity, we divide the remainder 
— lac — 2ftc-)-c* by twice this root, and obtain — c, which we 
write both in the root and in the divisor. If this competed divisor 
is multiplied by — c, and the product subtracted from the dividend, 
nothing remains. 



150 ELEMENTARY ALGEBRA. 

Hence, to extract the square root of a polynomial, 

RULE. 

Arrange the terms according to the powers of some letter. 

Find the square root of the first term, and write it as the 
first term of the root, and subtract its square from the given 
polynomial. 

Divide the remainder by double the root already found, 
and annex the result both to the root and to the divisor. 

Multiply the corrected divisor by this last term of the root, 
and subtract the product from the last remainder. Proceed 
as before with the remainder, if there is any. 

2. Find the square root of 4 .<;^ — 4 a:^^ -[" y*. 

Ans. 2x — y^. 

3. Find the square root of a^ -\-2ab-\-P-\-4:ac 
+ 4 5 c + 4 c2. Ans. a + h + 2c. 

4. Find the square root of 9 x^ — 12 x^ -{- 4=x^ -{- 6 ax^ 

— 4:ax-\-a^. Ans. Sx'^ — 2x-\-a. 

5. Find the square root of4a^ -\- Sab — 4a -|~ ^^^ 

— 46+1 Ans. 2a + 2b—l. 

6. Find the square root of 25 x^ — lOx^ -{- Gx"^ — x 
-\- {. Ans. 5^2 — oc -\- ^. 

1. Find the square root of a?^ + 2 x^ — x^ — 2x^ -{- x^. 

8. Find the square root of 4«^ — 4:ab -\- b^ — 4ac 
^4.ad-[-2bc-\-2bd-\-c''-^2cd-[-d\ 

Ans. 2 a — b — c — d. 

9. Find the square root of x^ — 4a?^ -[" ^^^ — ^^* 
J^bx'' — 2x-{- 1. 

10. Find the square root of 4a4 -|- Sa^b — SaH^ 
^I2ab' + 9b\ 



F.VOLUTION. 161 

Note 1. — According to the principles of Art 136, the signs of 
the answers given above may all be changed, and still be correct 

Note 2. — No binomial can be a perfect square. For the square 
of a monomial is a monomial, and the scjuare of the polynomial with 
the least number of terms, that is, of a binomial, is a trinomial. 

Note 3. — A trinomial is a perfect square when two of its terms 
are perfect squares and the remaining term is equal to twice the 
product of their square roots. For, 

(a-|-6)» = a'-f 2a6 + 6» 
(a — 6/ = a' — 2a& + 6» 

Therefore the square root of a' ± 2 a 6 -[- i' is a ± 6. Hence, to 
obtain the square root of a trinomial which is a perfect square, 

Omitting the term that is equal to twice the product of the square 
roots of the other two^ connect the square roots of the other two by the 
tign of the term omitted. 

1 1 . Find the square root of / + — * 

4 A 4 

Ans. 5-~ 

12. Find the square root of x' -f- 2x + 1. 

Ans. X -\-\. 

13. Find the square root of 4a:^ — 8xy -|- 4y^. 

14. Find the square root of ^- — 2ab -\- ^h^. 

15. Find the square root of 16y^ + 40yr* + 25 2*. 

Note. — By the rule for extracting the square root, any root whose 
inde.x is any power of 2 can be obtained by successive extractions 
of the square root Thus, the fourth root is the sqtiare root of the 
square root ; the eighth root is the square root of the square root of 
the square root; and so on. 

16. Find the fourth root of a« — I2a«6 + 54a*6» 
— 108o^» + 8l6^ Ans. a" — Zb, 



152 ELEMENTARY ALGEBRA. 



IT. Find the fourth root of -, + -1 + -^. 4- — 4- - 



Ans. -A 



X 



18. Find the fourth root of x^ — 4.x' + 10a;« — - 16a:« 
J^lQx^—Ux^-\-\{)x^ — 4.x-\-l. Ans. t^^x+I. 

146t To find any root of a polynomial. 

Since, according to the Binomial Theorem, when the terms of a 
power are arranged according to the power of some letter begin- 
ning with its highest power, the first term contains the first term 
of the root raised to the given power, therefore, if we take the re- 
quired root of the first term, we shall have the first term of the root. 
And since the second term of the power contains the second term of 
the root multiplied by the ne^jt inferior power of the first term of the 
root with a coefficient equal to the index of the root, therefore if we 
divide the second term of the power by the first term of the root raised 
to the next inferior power with a coefficient equal to the index of the 
root, we shall have the second term of the root. In accordance with 
these principles, to find any root of a polynomial we have the following 

RULE. 

Arrange the terms according to the powers of some letter. 

Find the required root of the first term, and write it as 
the first term of the root. 

Divide the second term of the polynomial by the first term 
of the root raised to the next inferior power and multiplied 
by the index of the root. 

Involve the whole of the root thus found to the given power, 
and subtract it from the polynomial. 

If there is any remainder, divide its first term by the di- 
visor first found, and the quotient will be the third term of 
the root. 

Proceed in this manner till the power obtained by involv- 
ing the root is equal to the given polynomial. 



EVOLUTION. 153 

Note 1. — This rule verifies itself. For the root, whenever a new 
term is added to it, is involved to the given power, and whenever the 
root thus involved is equal to the given polynomial, it is evident that 
the required root is found. 

Note 2. — As powers and roots are correlative words, we have 
used the phrase given power, meaning the power whose index is equal 
to the index of the required root, and the phrase next inferior power 
meaning that power whose index is one less than the index of the 
required root. 

1. Find the cube root of a« — 3 a'^ + 5 a* — 3 a — 1. 

OPERATION. 

Constant divisor, 3 a*) a' — 3 a^ -|- 5 a* — 3 a — 1 (a^ — a — 1 

gg _ 3 flS -I- 3 q^ _ gg 

— 3 a*, Ist term of remainder. 



a«_.3a» + 6a« — 3a— 1 

The first term of the root is a*, the cube root of a*, a* raised 
to the next inferior power, i. e. to the second power, with the co- 
efficient 3, the index of the root, gives 3 a*, which is the constant 
divisor. — 3 a*, the second term of the polynomial, divided by 

3 o*, gives — a, the second terra of the root, (a* — a)' = a* — 3 a* 
4" 3 a* — a* ; and subtracting this from the polynomial, we have — 3 a* 
as the first term of the remainder. — 3 a* divided by 3 a* gives 

— 1, the third term of the root, (a' — a — 1)'= the given poly- 
nomial, and therefore the correct root has been found. 

2. Find the fourth root of 16 x* — 32 ar« y* + 24 ar* y* 

— 8ar/ + y". 

OPERATION. 

4 X ( 2 x) • =* 3 2 a:") 1 6 a:* — 3 2 z" y» -f- 24 x« y* — 8 a; / -f y» ( 2 z — y» 

16 a;* — 82a:»y»-|-24a:'y* — 8xy'-}-y' 

3. Find the cube root of a» + 3 a" 6 + 3 a ft» + 6^ — 3 a^ c 
_>.6a6c — 36='c + 3ac« + 3ir»— c«. 

4. Find the fourth root of 16 a* c* — 32 a» c» x + 24 a« (r* ar» 

— 8acx^ + x*. 

7* 



154 ELEMENTARY ALGEBRA. 

SECTION XVII. 

RADICALS. 

147. A Radical is the indicated root of any quantity, 
as \/ X, d^ , \/2, 3^, &c. 

148. In distinction from radicals, other quantities are 
called rational quantities. 

149. The factor standing before the radical is the co- 
efficient of the radical. Thus, 2 is the coefficient of \/2 
in the expression 2 \/ 2. 

150. Similar Radicals are those which have the same 
quantity under the same radical sign. Thus, \/a, 2 \/a, 
and X h/a are similar radicals ; but 2 \/a and 2 \^b, or 
2 x^ and 2 x* are dissimilar radicals. 

151. A Surd is a quantity whose indicated root cannot 
be found. Thus, \/2 is a surd. 

The various operations in radicals are presented under 
the following cases. 

CASE I. 

152. To reduce a radical to its simplest form. ^ 

Note. — A radical is in its simplest form when it contains no 
factor whose indicated root can be found. 



1. Reduce \/^5a^b to its simplest form. 

OPERATION. 



^15aH = A/25a^ X Sb=z^25a^xV^h = 5a\/Sb 

We first resolve 75 a' b into two factors, one of which, 25 a', is 
the greatest perfect square which it contains ; then, as the root of 



RADICALS. 155 

the product is equal to the product of the roota (Art 143, Note 2)» 
we extract the square root of the perfect square 25 a", and annex 
to this root the factor remaining under the radical. Hence, 

RULE. 
Hesolve the quantity under the radical sign into two fac- 
tors, one of which is the greatest perfect power of the same 
name as the root. Extract the root of the perfect poicer, 
multiply it by the coefficient of the radical, if it has any, and 
annex to the result the other factor, with the radical sign 
between them. 

Reduce the following expressions to their simplest 
form : — 



2. 
3. 


^/12a:. 




9a« 


v/' 


3c 

v/1 


Ans. 2\/3x. 
Ans. 7 x' i^~x. 


4. 


^72a»6». 


Ans. 2a^96». 


6. 


b^64:ab\ 


Ans. 10 6/^^4 a. 


6. 

1. 


SA^U1aH\ 
25 \/ 56 a:. 


Ans. 21 a 6=^3. 

Ans. 60^7x. 


8. 
9. 


4x/128r«y. 
-C/343x». 




10. 


/ 27a«c 
Y l2S2*y 




11. 
12. 


1 1l€?C 

V 128x*y- 

Y 64 2 
>v/16r^y» — 32 






Vl6x«y«. 


— 32: 


rV 


Gx^y^Vl — 2ar'y« 



= 4a:y \/l — 2x2yS Ans. 



156 ELEMENTARY ALGEBRA. 



13. 4-^81a«c + 27a«. Ans. 12a4^3c + l. 

14. (a + h) \/3a^ — Qab-\- Sb\ Ans. (a" — b^) ^"3. 



15. 1 ^250x^f—12ox^y\ 

16. (a: — y) (a^ar — a^y)*. 

18. \/^^^16. 



-v/— 16 = \/16 V— 1=4:\/— 1, Ans. 



19. 4/ —1250. 



20. Vl9«' — 4^2 

153. When a fraction is under the radical sign, it can 
be transformed so as to have only an integral quantity 
under the radical sign, by multiplying both terms of the 
fraction by that quantity which will make its denominator a 
perfect power of the same name as the root, and then re- 
moving a factor according to the Bule in Art. 152. 



1. Reduce iy to its simplest form. 



OPERATION. 



<j1 = ^li = slh^~^=\'^^ 

Transform each of the following expressions so as to 
have only an integral quantity under the radical sign. 

2. iy/|- Ans. i v'^e. 

3. 4^|. Ans. *^^9. 

4. ?y/]. Ans. 1^^343. 
5- Isjm^- Ans. i>/347. 



RADICALS. 16T 









le- Ana. 1^30. 



10. .vi51 

*' 1876 



11. (a + 6)y/^. Ans. V^-6». 

CASE II. 
151* To reduce a rational quantity to the form of a 
radical. 

1. Reduce 3 a:* to the form of the cube root. 

OPERATION. Since 3 a:* is to be placed 

3 2^ __ 3^ 27 a:* under the form of the cube 

root without changing its 
value, we cube it and then place the radical sign, ^, over it It is 
evident that ^ 21 z? = 3 a:". Hence, 

RULE. 
Involve (he quantity to the power denoted by the index of 
the root required, and ph/ce the corresponding radical sign 
over the power thus produced. 

2. Reduce 4 a* 6 to the form of the square root. 



Ans. A^lQa^i^. 
3. Reduce 2ab'^c~^ to the form of the fifth root. 



Ans. 4/32a'b^''c'^. 



4. Reduce ^a'c^ to the form of the cube root. 

9 



168 ELEMENTARY ALGEBRA. 

5. Reduce -z — r to the form of the fourth root. 

Bxyt 

6. Reduce x — 2i/ to the form of the square root. 



Ans. \/ x^ — 41x1/ -\- 4: y^. 

155% On the same principle the rational coefiScient of a 
radical can be placed under the radical sign, by involv- 
ing the coefficient to a power of the same name as the root 
indicated by the radical sign, multiplying it by the radical 
quantity, and placing the given radical sign over the product. 

1. Place the coefficient of hi^ly under the radical 
sign. 

OPERATION. 



6 A^ 2y = >^ 125 >ij/ 2^^ = ^ 250y 

In the following examples, place the coefficient under 
the radical sign. 



2. 3>^4a;8y. Ans. ><y324x'y. 



3. 2xyf^2x'y. Ans. ^Ua^y^. 

4. f x/i. 

5. ^\/l4. 



6. {a — h) ^^^' Ans. ^a^ —■2aH + ab"", 

T. 4.xy\^l — 2x^f. 

CASE III. 

156. To reduce radicals having different indices to 
equivalent ones having a common index. 

1. Reduce \/« and ^ b to equivalent radicals having 
a common index. 



RADICALS. 169 

OPERATION. In thirf case we write the radicals 

4 I B/ — ii witli their fractional indices : and 

a^ =: a^ =: /^ a . . 

then, as the denominator is the in- 

b^ = b^ =z A^ h^ dex of the root, in order that the 

two radicals may have the same 
root-index, we reduce the fractional indices to equivalent ones hav- 
ing a common denominator. It is evident that we have not changed 
the values of the given radicals by the process. Hence, 

RULE. 
Reduce the fractional indices to equivalent ones having a 
common denominator ; involve each quantity to the power de- 
noted by the numerator of the reduced index ^ and indicate 
tlie root denoted by the denominator, 

2. Reduce \^2 and ^^3 to equivalent radicals having 
a common index. 



3* = 3^^ = -^3^ == ^27 ) 



Ans. 



3. Reduce \/ | and v^ ^ to equivalent radicals having 
a common index. Ans. ^j^^ and /^ r^. 

4. Reduce 4/ - and 1/ | to equivalent radicals having 
a common index. 



5. Reduce ^ a, ^ a — b, and \/ a -{- b to equivalent 
radicals having a common index. 

Ans. v' < v' (^i^^y, and iy (a~+by. 

6. Reduce \/2, \/4, and /^ 3 to equivalent radicals 
having a common index. 

7. Reduce \/ x and v^y to equivalent radicals having 
a common index. Ans. \/x^ and \/y". 



160 ELEMENTARY ALGEBRA. 

CASE IV. 
157. To add radical quantities. 

1. Add \/ic and Vy. Ans. \/« + Vy^ 
It is evident that the addition can only be expressed. 

2. Add Zf>/x and b\fx. Ans. ^ s/x. 

It is evident that 3 times the ^ x and 6 times the ^~x make 8 

times the y/a;. 

3. Add V'S and \/ 50 together. 

OPERATION. In this case we make the radi- 

f^~~^ = 2 \/^ ^^^ ipa-rts similar by reducing them 

.-rx c /"H- *o t^®"* simplest form (Art. 152), 

^ "^^ — — — and then add their coefficients as 

Sum = T \/ 2 in Example 2. Hence, 

RULE. 

Make the radical parts similar when they are not, and 
prefix the sum of the coefficients to the common radical. If 
the radical parts are not and cannot he made similar, con- 
nect the quantities with their proper signs. 



4. Add 2\/60«a; and 3\/98aa;. Ans. 31\/2aic. 



6. Add 4>^24ar8 and x^^l. Ans. 110:^^3. 

6. Add \/2T and \/^363. Ans. 14\/'3. 



Y. Add -^512a:* and .^162y*. 

Ans. {4.x -{-Zy) 4^2. 

8. Add ^1) and ^/\ 

VT=\/^\/5 = i\/5; V5 + i\/5 = f\/"5, Ana. 

9. Add ^^ and ^1|^. Ans. ^ ^l2 . 



RADICALS. 161 

10. Add Vl, 10^7f» and 6/v/20. Ana. 13 V^. 

11. Add Vl^ and a/2^. 

CASE V. 
158i To subtract one radical from another. 

1. From \/T5 take \/'2T. 

OPERATION. We make the radical parts sim- 

.-=T r /-q ilar by reducing them to their sim- 

~~ plest form (Art. 152). And 3 y/l 

taken from 5 y^ 3 evidently leaves 
2 v' 3. Hence, 



V27 = 3^/3 
2v^3 



RULE. 
Make the radical parts similar when they are not, sub- 
tract the coefficient of the subtrahend from that of the min- 
uend, and prefix the difference to the common radical. If 
the radical parts are not and cannot be made similar, indi- 
cate the svbtracHon by connecting them with the proper sign. 
__ _ • 

2. From ^^ take >^ 3. Ans. 2-^3. 

3. From 9 s/ a^xy^ take 3 a \/ x^*. 

Au8. ^ayjs/x. 



4. From T \/20x take 4\/45a:. Ans. 2\/5ar. 

6. From /^ 500 take ^T08. Ans. 2 ^1^, 

6. Prom 2^^^ take \/^. Ans. ttV^- 

7. From >v/ | take \/^. Ans. ;^ V^O. 



8. From 2^n6a:» take .^y891x». 

9. From a 4/1^ take 7 *i/a^3^. 



10. From >C^ 1174 take ^1892. 



162 ELEMENTARY ALGEBRA. 

CASE VI. 
159. To multiply radicals. 

1. Multiply 3 V « by 5 f^l. 

OPERATION. 

3 s/~a X 5 \/T = 3 X 5 X V « X V^ = 15 s/~ab 

As it makes no diflference in what order the factors are taken, 
we unite in one product the numerical coefficients ; and >^ ay^^h 
= \/~ab (Art. 143, Note 2). 



2. Multiply 4\/2a6 by 5 /v^3«y. 

We reduce the radical 
OPERATION. . , . , 
parts to equivalent radicals 

4v2a6= 4Y^ 8a o having a common index 

(Art. 

ply as 

ample. 



5^Sax= 6^ 9a^x^ (Art. 156), and then multi- 

Product = 20 ^W^W ^^y ^' ^^ *^' preceding ex- 



3. Multiply ^a by \/ a. 



OPERATION. 



a^ X a2 zz= >^a'^ X a^ = \/a\ or a* 

Multiplying as in the preceding examples, we have ^ a*, or a^ ; 
but I = ^ -|- ^ ; i. e. the index of the product is the sum of the 
indices of the factors. 

From these examples we deduce the following 

RULE. 

I. Reduce the radical paints, if necessary, to equivalent 
radicals having a common index, and to the product of the 
radical parts placed under the common radical sign prefix 
the product of their coefficients. 

II. Boots of the same quantify are multiplied together by 
adding their fractional indices. 





RADICALS. 


163 


4. 


Multiply 3VT0by 4V'5. 


. Ans. 60 V 2. 


6. 


Multiply 4 -C^ a ar* by a Vx. 


Ana. ^axA^/d^x. 


6. 


Multiply oa/c by b^/c. 


Ads. a&c. 


1. 


Multiply \/ x" by ^ x. 


Ans. x«. 


8. 


Multiply ^fhyV^. 


Ans. 1*, or ^16807. 


9. 


Multiply \^x by v^x. 


Ans. V^""^". 



10. Multiply 2 /v/a + 6 by 6x^a + 6. 

Ans. 12 x-^ (« + *)*. 

11. Multiply /k/x, *yx, and /^x together. Ans. x^*. 

12. Multiply 3 a{/^ by 2 V 8. Ans. 6 >C/2. 



13. Multiply a\/x-^y by h mJ xy. Ans. aJy. 

14. Multiply (a + ft)i by (a — 5)^ Ans. (a^ — J^jl 
16. Multiply i \/T by 3 \/ 1 

16. Multiply 2 V'S by 4 V"8. 

CASjE VII. 
160* To divide radicals. 



1. Divide 60\/15x by 4\/6x. 

OPERATION. As division ia finding 

60 \/T5x -i- 4 i^~hx= 15 V 3" * quotient which, multi- 

plied by the divisor, will 
produce the dividend, the coefficient of the quotient must be a 
number which, multiplied by 4, will give 60, the coefficient of the 
dividend, i. e. 15; and the radical part of the quotient must be a 
quantity which, multiplied by y^Sx, will give v^lSa:, i.e. v^3; the 
quotient required, therefore, is 15 v' 3. 



164 ELEMENTARY ALGEBRA. 

2. Divide 6 M^Ty by 2 .^ 2y. 



OPERATION. 



We reduce the radical parts to equivalent radicals having a com- 
mon index (Art. 155), and then divide as in the preceding example. 

3. Divide \/a by /^ a. 

OPERATION. 

^ ^ „i = ^a^^Zr^2 _ ^- or J 

Dividing as in the preceding examples, we have ^~a, or a^. But 
^ = i — i ; i' e. the index of the quotient is the index of the divi- 
dend minus the index of the divisor. 

From these examples we deduce the following 

RULE. 

I. Reduce the radical parts, if necessary, to equivalent 
radicals having a common index, and to the quotient of the 
radical parts placed under the common radical sign prefix 
the quotient of their coeficients. 

II. Roots of the same quantity are divided by subtracting 
the fractional index of the divisor from that of the dividend. 

4. Divide 16 a/ ax by S\/a^x. Ans. 2\/«~^. 



6. Divide 4:A^d' — b"^ by 2 \/« — *• 

Ans. 2A/a -\- b. 

6. Divide Q\/'21 hj Sa/S. Ans. 6. 



T. Divide *>/ x by \/x. Ans. v^"*""- 

8. Divide \/a by Xl/b. Ans, i/r- 

9. Divide 3 by \^~3. Ans. -v/'S. 



RADICALS. 165 

10. Divide x by ^x. Ans. ^x^. 

11. Divide 4a«\/x by 2cr^\/~y, Ans. ^^^Jy 

12. Divide V 6" by ^5^ 

13. Divide 4^1 by </T. 

14. Divide /^a by f^ a. 

15. Divide I ^1 by I ^f. 

CASE VIII. 
161 1 To involve radicals. 

1. Find the cube oi Z^x. 

OPERATION. 

(3V^)»==3\/x X Zs/xX 3\/7 

In accordance with the definition of involution, we take the quan- 
tity three times as a factor. By Art. 159 the product is 27 y/ a:*. 

2. Find the square of 2 .^ a. 

OPERATION. ^° ^^^ ^^® ^^ ^^^^ "^^^ 

_ 4 1 *^® fractional exponent, and 

(2 aJ/ a)* = (2 a*)* = 4 a* found the square of the given 

quantity by multiplying its 
exponent by the index of the required power, according to Art. 1 26. 
Hence, 

RULE. 

I. Involve the radical as if it were rational, and placing 
it under its proper radical sign, prefix the required power 
of its coefficient. 

II. A radical can be involved by multiplying its fractional 
exponent by the index of the required power. 



166 ELEMENTARY ALGEBRA. 

Note. — Dividing the index of the root is the same as multiply- 
ing the fractional exponent. Thus the square of ^ a is i/~a. For 
(a^f = a^, or y' a. 

3. Find the cube of 3x\/ a. 

Ans. 21 a x^ a/ a, or 21 a^ x. 

X 2 

4. Find the square of 4 a^. Ans. 16 a^, or 16 ^a^. 

5. Find the fourth power of 3 V^. Ans. 81 x^. 

6. Find the nth power of a/^/x. Ans. dl't^l^. 
1. Find the fourth power of 6 a/^, Ans. 25. 

8. Find the cube of B \/~T. Ans. 189 VT. 

9. Find the fourth power of r. 

10. Find the cube of 2 a/Tx. 



CASE IX. 
162. To evolve radicals. 

1. Find the cube root of Sa^-^o^. 

OPERATION. As the root of the product is 

' * ^ (Art. 143, Note 2), we prefix to 

the cube root of the radical part the cube root of the rational part. 
The cube root of the radical part must be a quantity which, taken 
three times as a factor, will produce &a^3?\ i. e. ^ax. 

2. Find the fourth root of ^ x. 



— — f \\\ i_ the fractional exponent, and 

V (^a: = (a;*) = x^, or '^ x found the fourth root by di- 

viding the exponent of the 
given quantity by the index of the required root, according to 
Art 143. Hence, 



RADICALS. 167 

RULE. 

I. Evolve the radical as if it were rational, and, placing 
it under its proper radical sign, prefix the required root of 
its coefficient. 

II. A radical can he evolved by dividing its fractional 
exponent by the index of the required root. 

Note. — Multiplying the index of the root is the same as divid- 
ing the fractional exponent Thus, the square root of ^ a is ^ a. 

For (a^)^ = ai, or ^ a. 

3. Find the square root of ba/^A^x. 

(5 a A^Ti)^ = (^SOOa'ar)^ =>^500a«x, Ans. 

4. Find the cube root of x'^ ^ a^ b. Ans. 1/ -W"* 
6. Find the fifth root of ar^\/^. Ans. \/x. 
6. Find the fourth root of i VT- Ans. ^'^. 
1. Find the cube root of 1 a,/S. Ans. ^TlT. 
8. Find the square root of 12 /^b. 

POLYNOMIALS HAVING RADICAL TERMS. 

163. It appears from the principles already established, 
that the laws which apply to calculations with quantities 
which have exponents, apply equally well whether the 
exponents are positive or negative, integral or fractional. 
The following examples, therefore, can be done by rules 
already given. 

1. Add 4a — 3 \/y and 3 a + 2 \/y. Ans. la — Vy. 



2. Add 3 X + >^ 135 and 1 x — ^ 1080. _ 

Ans. lOar — 3>^6. 



168 ELEMENTARY ALGEBRA. 

3. Add 2 V 28 — V 27 and 2 V 63 + \/i8. 



4. Subtract 15 a; — \/50a from 13 a: — s/^a. 

Ans. 3\/^ — 2ar. 

5. Subtract f^ as? — \/4 6 from ^J ax — \/16 6. 

Ans. A^ ax — Xfs/a — 2\/6. 

6. Subtract .i^"32 — ^"242 from — 3 f^l — T \/"3. 
T. Multiply /v/a — \/^ by \/a — /\/a;. 

OPERATION. 

\/a — fn/ X 



a — ^ ah — /^ ax -\- /^ bx 

8. Multiply xy -\- \^ ah by 4 — s/ ah, 

Ans. ^xy -\- (4 — a; 3/) /y/o^ — ah. 

9. Multiply T + \/T0 by 6 — \/l0. 

Ans. 32 — \/T0. 

10. Multiply \/a -j- V^ by \/« — V^- -^i^s. a — J. 

11. Multiply V'5 — 4-^"3 by ^45 + /^"O. 

12. Multiply ^ VI + T \/"3 by i Vl^ — T V 3. 

13. YiWi^Q s/ ax -\- M^ ay -\-x-\- s/ xy \y^ s/ a-\' s/ X. 

OPERATION. 

\/a -j- \/a;) is/ ax -\- ^ ay -\- x -\- s/ xy (\/ar -f- \/y 
^/ ax -f- a: 

V^ + V^ 

s/ ay +V^ 



RADICALS. Ig9 

14. Divide V^ — V^ — V^ + \/^ by v^ — V^. 

Ans. s/~ot. — \/^. 

16. Divide <^ x-\-^x-\-^y^-\-^y^ bya:+y* 

16. Divide a: — y by V^ — Vy. Ans. V^ + V.y. 

17. Divide4xy + 4>v/^--ary>v/^ — a*by4 — V"^. 

18. Expand (V^ + \/~yy. Ans. x + 2 V^ + y. 

19. Expand (a^ — IT^y. Ans. a — 2 4/?^+ 1. 

20. Expand (^ — ^1)*. 

Ans. o^ — 4a*6i + 6a6 — 40*6' 4- ja 

21. Expand (4 — V^)». Ans. 100 — 61^3. 

22. Expand (a-i — ar-i)». 

Ans. a-l — 3 a-^ x-^ + 3 a-4a:-« — x-*. 

?3. Expand (l-^ly. 

Ans. l_-4. + i. ?_ I i. 

24. Expand (^|--^|y. 

Ans. f?-i4j^+:.y_l^ , ^ 

25. Find the square root of a — 2 a^ i^ -j- i^. 

Ans. t^~a — /^~W, 

26. Find the cube root of x* — 3 x^y^ + 3 ary^ — y. 

Ans. X — ys. 

27. Find the fourth root of 16 a — 32 a^y* + ^4 ar^yl 
-8xV + y^. Ans. 2 a^~yf 



170 ELEMENTARY ALGEBRA. 



SECTION XVIII. 

PURE EQUATIONS 

WHICH REQUIRE IN THEIR REDUCTION EITHER INVO- 
LUTION OR EVOLUTION. 

164. A Pure Equation is one that coDtains but one 
power of the unknown quantity ; as, 

A^x -\- ac = h, 4 aj2 -|- 3 = T, or 14 a;" = a 5. 

165. A Pure Quadratic Equation is one that contains 
only the second power of the unknown quantity ; as, 

6x'^'-14:a=5lb, ay^=lScd, ov acz'^ 14. 

166. Radical Equations, i. e. equations containing the 
unknown quantity under the radical sign, require Invo- 
lution in their reduction. 

167. To reduce radical equations. 

1. Reduce \/ a; — 3»=^. 

operation. 
V"^ — 3 = 8 
Transposing, a/ x =^ 11 

Squaring, x = 121 



2. Reduce /^x — 4 + 7 = 10. 



operation. 
^ 1^4 _|_ 7 == 10 



Transposing, ^x — 4 = 3 

/Cubing, as -^ 4 == 2t 

Transposing, *c =? 31 



RADICAL EQUATIONS. * 171 



3. Reduce ^^^'±^~^ 
y/a 


= v<.. 




OPERATIOM. 






Clearing of fractions, 


V/df2 + Va:= a 


Squaring", 


d^ + A^x= a" 


Transposing, 


^x=z a^ — d^ 


Squaring, 


a: = (a^ — d'f 



Hence, to reduce radical equations, we deduce from 
these examples the following general 

RULE. 
Transpose the terms so that a radical part shall stand by 
itself; then involve each member of the equation to a power 
of the same name as the root ; if the unknown quantity is 
still under the radical sign, transpose and involve as before ; 
finally reduce as usual. 

4. Reduce 4 + J + 3 \/lc = ^J-. Ans. x=l6. 

5. Reduce - 4/ - = - . Ans. x = 2. 

6. Reduce (>v/7+ 4)* = 2. Ans. x = 144. 



T. Reduce V 11 + arm \/x + 1. Ans. a; = 25. 



8 Reduce ^x — 7 == a^ x -f 18 — VS. 

Ans. ar = 27. 

9. Reduce ■ ^^ = -^ . Ans. x = 

X ex yj X 1 <? 

10. Reduce ^- ~^ = .V^^nJL . Ans. x = 9. 

V^z-f-10 v/x-f28 



172 ELEMENTARY ALGEBRA. 



11. Reduce s^ x -\- a^ x — a= . 

SJ X — a 

12. Reduce Vx — 30 + \^ x -j- 21 = \/ x — 19. 



13. Reduce \/9^+ 13 = 3\/^+ 1. Ans. a; r= 4. 

14. Reduce V — =^ ^, ' Ans. a? = 5. 

y/Sx-j- 1 v^5x-j- 2 



15. Reduce \/^"^ — 32 = ic — ^V 32. 

168i Equations containing the unknown quantity in- 
volved to any power require Evolution in their reduction. 

169. To reduce pure equations containing the unknown 
quantity involved to any power. 

1. Reduce --_-=-. 

OPERATION. 

4a^ 3 97 Clearing the given equation of 

5 7 35 fractions, transposing, and divid- 

28a;'^ — 15= 97 ing, we have x^ == 4; extract- 

28 ic^ = 112 ing the square root of each mem- 

^2__ ^ ber of this equation, we have 

x=±2 x=±2. (Art. 136.) 



2. Reduce 7 a;« — 89 = 100. 



OPERATION. 

7 a:» — 89 = 100 

7a:«=189 

ar«= 27 



Transposing and dividing, we 
have 3? = 27; extracting the 
cube root of each member of 
this equation, we have a: = 3 



X := 3 Hence, 

RULE. 
Reduce the equation so as to have as one member the un- 
known quantity involved to any degree, and then extract that 
root of each member which is of the same name as the 
power of the unknown quantity. 



PURE EQUATIONS ABOVE THE FIRST DEGREE. 173 

Note. — It appears from the solution of Example 1 that every 
pure quadratic equation has two roots numerically the samej but with op- 
posite signs, 

3. Reduce-5ar»+ T==?a:^ + 3. Ans. x = ± 6. 

X^ x^ 

4. Reduce a = i — ^ • 

c a 



. , bed — acd 

Ans. X = 









■^V c-d 


5. 


Reduce = 10. 






Ans. x= ± 4. 


6. 


Reduce 3x« + ^ = i. 




Ans. 0: = ^-. 


7. 


Reduce ^^ + 60 = 1. 




Ans. a: = — T. 


8. 


^ , X — 4 2r— 1 
Reduce ^^_^,= ^_^^' 


Ans. 


x=±V-5. 


9. 


Reduce 4a:' — 4a:» = 0. 







10. Reduce 6ar»— 3a: =3x2 — 32: + 60. 

1 7 

1 1 . Reduce x 4- - = :5 1. 

12. Reduce 2x + 2= (x+ If. 

13. Reduce I -\- Ux-^=2 ^ 2x-\ 

14. Reduce Sar^ — 6ar« = 2x-2 + Sx"^ — J. 
16. Reduce (c + x)» — 6 c2x= (c — x)»+ 16c*. 
iz>r>j 85 8z — 3 8x4-3 

16. Reduce ,^ — - — r— = - — ^. 

42 3x-}- 3 3x — 3 

IT T> J a:«-f-x4-8 , x» 4- x — 8 „ 

17. Reduce -^^-f- + -J—^- = 2. 



174 ELEMENTARY ALGEBRA. 

170i Equations containing radical quantities may re- 
quire in their reduction both Involution and Evolution; 
and in this case the rule in Art. 167, as well as that in 
Art. 169, must be applied. Which rule is first to be ap- 
plied depends upon whether the expression containing the 
unknown quantity is evolved or involved. 



1. Reduce 17 — \/x« — 2 = 12. 

OPERATION. 



IT — Va:^ — 2 = 12 



Transposing, &c., s/ x^ — 2 = 5 

Squaring, a:^ — 2 = 25 

Transposing and uniting, ^ = 27 

Extracting the cube root, a: = 3 



2. Reduce (V^c' — 4 + 3)^ = 125. Ans. a; = 2. 



3. Reduce i/ — =z a^ x. Ans. x-=z ± - 

y 2a; .2 

a-\-h 



4. Reduce \f x -\- a =. 



\ X — a 



Ans. x= ± A^2a^-{-2ab + b\ 



5. Reduce f- ^3 (.^ + ii)^^ 



6. deduce Ai^2x^ + Sx^ + 24.x^+S2x=zx + 2. . 

7. Reduce 4^9 {x' + 19) + 100 — ^ = ^^ • 

171 1 Equations which contain two or more unknown 
quantities may require for their reduction involution, or 
evolution, or both. In these equations the elimination is 
effected by the same principles as in simple equations. 
(Arts. 112-114.) 

3^ 

1. Given -^5 4 "^^ [>-, to find x and y. 
2x^ + , 



^ — 14 ? 
y = 54) 



PURE EQUATIONS . 


ABOVE THE FIRST DEGREE. 






OPERATION. 




^-H^* 


(1) 


2x' + y= 54 


(2) 






r ^-^« 


(3) 






r=iio 


(4) 


15-?=14 


C) 


a^= 25 


(5) 


y= 4 


(8) 


X =±5 


(6) 



176 



Subtracting four times (1) from (2), we obtain (4), which re- 
duced gives (6), or X = ± 5 ; substituting this value of x in (1), 
we obtain (7), which reduced gives (8), or y = 4. 

Find the value of the unknown quantities in the follow- 
ing equations: — 

2. Given \'^=''-^l. Ans.l^=±^- 

3. Given |'"-'^='^|. Ans. \-=±^- 

Ma:«2 = 20 \ ra:== ± 1. 

4. Given ^ 2 x ar = 10 >• . Ans. -j y = ± 3. 

(3y.z = 45) (2:= ±5. 

6. Given K^-^^=n. Ans. |- = 2J- 

6. Given |x« + y = 97 > 

(x — y = 1/ — 2x> 



7. Given 



(x*--2y«=14 ) 



176 ELEMENTARY ALGEBRA. 



PROBLEMS 

PRODUCING PURE EQUATIONS ABOVE THE FIRST 
DEGREE. 

172. Though the numerical negative values obtained in 
solving the following Problems satisfy the equations 
formed in accordance v^ith the given conditions, they are 
practically inadmissible, and are therefore not given in 
the answers. 

1. A gentleman being asked how many dollars he had 
in his purse, replied, " If you add 21 to the number and 
subtract 4 from the square root of the sum, the remainder 
will be 6.^' How many had he? 

SOLUTION. 

Let X = number of dollars. 



Then, ^/x + 2i—4.= 6 
Transposing, /\/ x -{- 21 =. 10 

Squaring, a; + 21 = 100 

Transposing, x= 19, number of dollars. 

2. Divide 20 into two parts whose cubes shall be in 
the proportion of 27 to 8. Ans. 12 and 8. 

3. What two numbers are those whose sum is to the 
less as 8 : 3, and the sum of whose squares is 136? 

Ans. 10 and 6. 

4. What number is that whose half multiplied by its 
third gives 54 ? 

6. What number is that whose fourth and seventh 
multiplied together gives 46f ? Ans. 36. 

6. There is a rectangular field containing 4 acres whose 
length is to its breadth as 8:5. What is its length and 
breadth ? 



PURE EQUATIONS ABOVE THE FIRST DEGREE. 177 

V. There are two numbers whose sum is IT, and the 
less divided by the greater is to the greater divided by 
the less as 64 : 81. What are the numbers? 

Aus. 8 and 9. 

8. The sum of the squares of two numbers is 65, and 
the difl'erence of their squares 33. What are the numbers ? 

9. The sum of the squares of two quantities is a, and 
the difference of their squares b. What are the quantities ? 

Ans. ± VTTM^) a"d ± Vi la~^h). 

10. A gentleman sold two fields which together con- 
tained 240 acres. For each he received as many dollars 
an acre as there were acres in the field, and what he 
received for the larger was to what he received for the 
smaller as 49 : 25. What are the contents of each ? 

Ans. Larger, 140; smaller, 100 acres. 

11. What are the two quantities whose product js a 
and quotient hi . . — - , la 

Ans. ± V a ^ and ±1/7- 

12. What two numbers are as m : n, the sum of whose 

squares is a? ^ mi/a , . nd'a 

Ans. ± -, ^ and ± -; ^ 

Y/(m«-}-n«) V(m'-}-n«) 

13. What two numbers are as m : n, the difference of 
whose squares is a? , - ,- 

Ans. ± ^ and ± -, — 

14. Several gentlemen made an excursion, each taking 
$484. Each had as many servants as there were gentle- 
men, and the number of dollars which each had was four 
times the number of all the servants. How many gen- 
tlemen were there? Ans. 11. 

16. Find three numbers such that the product of the 
first and second is 12 ; of the second and third, 20 ; and 
the sum of the squares of the first and third, 34. 
8» L 



178 ELEMENTARY ALGEBRA. 

SECTION XIX. 

AFFECTED QUADRATIC EQUATIONS. 

173f An Affected Quadratic Equation is one that con- 
tains both the first and second powers of the unknown 
quantity ; as, 

Sx^ — 4: x=: 16; or ax — bx^ = c. 

174. Every affected quadratic equation can be reduced 
to the form 

x"^ -]- bx = c, 

in which h and c represent any quantities whatever, posi- 
tive or negative, integral or fractional. 

For all the terms containing a^ can be collected into one term 
whose coefficient we will represent by a ; all the terms containing x 
can be collected into one term whose coefficient we will represent 
by d; and all the other terms can be united, whose aggregate we 
will represent by e. Therefore every affected quadratic equation 
can be reduced to the form 



Dividing (1) by a, 

d e 

Letting - =b, and - = c, we have 

175. The first member of the equation x^ -\- bx = c 
cannot be a perfect square. (Art. 145, Note 2.) But 
we know that the square of a binomial is the square of 
the first term plus or minus twice the product of the two 
terms plus the square of the last term; and if we can 
find the third term which will make x"^ -{- bx a perfect 



aa^-\-dx = e 


(1) 


'a a 


(2) 


3^-{-bx = c 


(3) 



EQUATIONS OP THE SECOND DEGREE. 179 

square of a binomial, we can then reduce the equation 
x^ -\- bx = c. 

Since b x has in it as a factor the 8qi\are root of a:*, a:* can be 

the first term of the square of a binomial, and b x the second terra 

of the same square; and since the second term of the square is 

twice the product of the two terms of the binomial, the last term of 

the binomial must be the quotient arising from dividing the second 

term of the square by twice the square root of the first term of the 

square of the bi no- 

inial ; i. e. the last 
OPERATION. ^^ ^^ ^i^g ^j„^ 

x2-f-5ar = c (1) . , . bx b 

«" + ^^ + 4 = 4 + « (2) ^^^ therefore the 

J rn — third term of the 

x-\- -^=- ±. kI T '\~ ^ (^) square must be 

2 Y * b\ . 

— to each mem- 
4 

ber, we have (2), an equation whose first member is a perfect 

square. Extracting the square root of each member of (2), and 

transposing, we obtain (4), or x ^ — 9 "*■ v/ T "^ ^' ^^'^^ ** * 
general expression for the value of x in any equation in the form 
of z* -|" ^ ^ = <^* 

Hence, as every affected quadratic equation can be re- 
duced to the form x^ -\- hx^=. c, in which b and c repre- 
sent any quantities whatever, positive or negative, integral 
or fractional, every affected quadratic equation can be re- 
duced by the following 

RULE. 

Beduce the equation to the form x^ -\- bx=z c, and add 
to each member the square of half the coefficient of x. 

Extract the square root of each member, and then reduce 
as in simple equations. 



180 ELEMENTARY ALGEBRA. 

1. Reduce Ta;2 — 28 a; + 14 = 238. 

OPERATION. 

*lf — 2Sx-{- 14 = 238 
Transposing, T a;^ — 28 a: = 224 

Dividing by Y, x"^ — 4a;= 32 

Completing the square, x^ — 4 x -j- 4 = 36 
Evolving, X — 2 = ±6 

Transposing, a;:=:2±6 = 8, or — 4 

Note. — Since in reducing the general equation x^ -{- bx = c 
vre find x = — ^±i/j -f-c, every affected quadratic equation 
must have two roots ; one obtained by considering the expression 
2 ~\~ ^ positive, the other by considering this expression nega- 



/ b^ 
tive. Whenever 4 / - -j- c = these two roots will be equal. 



o -D J ^ rr , 13 x^ , X 

2. Reduce ^--- + -^ = - + -. 
operation. 

Clearing of fractions, 4a:^ — 2a: + 13 = lOa:^ -|- 5x 
Transposing, — Car* — 1x = — 13 
Dividing by — 6, x^ -\ = — 

Completing the square, ^' + i) +^^ = ^, + ~ =~ 

Evolving, a; + ^ = ± ^ 

Transposing, ^ = — — ± j^ ^\, or— 2^ 



EQUATIONS OF THE SECOND DEGREE. 181 

Note. — In completing the square, as the second term disappears 
when the root is extracted, we have written ( ) in place of it. 

3. Reduce 3x^ — 25 + 6x = 80. 

Ans. x=z 5, or — t. 

24 X 

4. Reduce x = 3. Ans. x=z6, or — 4. 

X 

5. Reduce2x + ^-±4=7. Ans. x = 2. 

' X 1 

Note. — In this example both roots are 2. 



x* 4- 4 

6. Reduce 1x ^^ =zbx — 1. 

X — 4 



Ans. X = 8, or — 1. 

t. Reduce 17 - ^-^-=^ == ^£^ + 10. 

Ans. X = 7, or — 27. 

8. Reduce T , + 3 = ^* A^^s. x = 10, or — 1 J. 



x+5 



9. Reduce j -^ = 4. 



,/v T> J 16 100 — 9x « 

10. Reduce —^ — = 3. 

x 4z" 



176. Whenever an equation has been reduced to the 
form x^ -\- bx = c, its roots can be written at once; for 

this equation reduced (Art. 175) gives x = — - ± t /- + c. 
Hence, 

TJie roots of an equation reduced to the form x^ -\- bx = c 
are equal to one half the coefficient of x with the opposite 
sign, plus or minus the square root of the sum of the square 
of one half this coefficient and the second member of the 
equation. 



182 ELEMENTARY ALGEBRA. 

In accordance with this, find the roots of x in the fol- 
lowing equations : — 

1. Reduce a:^ -|- 8 a: = 65. 

a; = — 4 ± /s/16 + 65 = 6, or — 13, Ans. 

2. Reduce a:2~ 10a; = — 24. 

x = b ± /v/25 — 24 = 6, or 4, Ans. 

3. Reduce x'^ — Qx=z — ^, Ans. a; = 5, or 1. • 

4. Reduce a:^ _|_ T a; = ITO. 

7 /^9 

^ = — 2 ± \/ X + ^"^^ = ^^' ^^~ ^'^' ^"«- 

6. Reduce a;^ + -x = -- 

^ = -i±\/^ + ^ = i or -1, Ans. 

1 9 

6. Reduce x^ + -a: =-• Ans. a: = 1, or — 1^. 

Y. Reduce a;2 — -zx^^ — — • Ans. a; = -, or -• 

5 25 5 5 

8. Reduce - = | -j- 5^. Ans. a: = 7, or — 5^. 



SECOND METHOD OF COMPLETING THE SQUARE. 

177. The method already given for completing the 
square can be used in all cases ; but it often leads to in- 
convenient fractions. The more difficult fractions are in- 
troduced by dividing the equation by the coefficient of 
x^y to reduce it to the form a;^-|-ia: = c. To present a 
method of completing the square without introducing these 
fractions, we will reduce equation (1) in Art. 174. 



EQUATIONS OF THE SECOND DEGREE. 183 
1. Reduce ax^ -\- dx = e. 

OPERATION. 

ax' + dx = e (1) 

a^a^-^-adx^ae (2) 

a'^x^ + adx + j=j + ae (3) 



ax 






X 



Multiplying (1) by a, the coefficient of a^, we obtain (2), in which 
the first term must be a perfect square. Since adx, the second 
terra, has in it as a factor the square root of a* r", a* 2* can be the 
first tenn of the square of a binomial, and adx the second term ; 
and since the second term of the square is twice the product of the 
two terms of the binomial, the last term of the binomial must be the 
second term of the square divided by twice the square root of the first 

term of the square of the binomial, or „ — = - ; and therefore the 

2 ax 2 

d* 
term required to complete the square is — , which is the square of 

d* 
one half of the coefficient of x in (1). Adding - to both members 

of (2), we obtain (3), whose first member is the square of a binomial. 
Extracting the square root of (3) and reducing, we obtain (5), or 



l{-Ht+-} 



Hence, to reduce an aflfected quadratic equation, we 
have this second 

RULE. 

Reduce the equation to the form ax^ -\- dx = e; then mul- 
tiply tlie equation by the coefficient of ar*, and add to each 
member the square of half the coefficient of x. 

Extract the square root of each member, and then reduce 
as in simple equations. 



184 ELE]VIENTARY ALGEBRA. 

Note 1. — This method does not introduce fractions into the equa- 
tion when the numerical part of the coefficient of x is even. When 
the coefficient of x* is unity, this method becomes the same as the 
first method. 

Note 2. — If the coefficient of 3? is already a perfect square the 
square can be completed without multiplying the equation, by add- 
ing to both members the square of the quotient arising from dividing 
the second term by twice the square root of the frst. This method 
also becomes the same as the first method when the coefficient of 
3^ is unity. 

Note 3. — As an even root of a negative quantity is impossible 
or imaginary, the sign of the first term, if it is not positive, must 
be made so by changing the signs of all the terms of the equation. 

2. Reduce 3 a:^ + 8 a; = 28. 

OPERATION. 

3x2_|_8a: = 28 
Completing the square, 9 a;^ + ( ) -f 16 = 16 + 84 = 100 
Extracting square root, 3a;-[-4=± 10 

Whence, 3a; = — 4 ±10 = 6, or — 14 

And x = 2, or — 4f 

3. Reduce 25ar» — 10 a; = 195. 

operation. 
25a;2_-i0a:=195 
Completing sq. by Note 2, 25a;2— () + 1 = 1 + 195 = 196 
Extracting square root, 5x — 1 = ± 14 

Whence, 5a:=l ± 14 = 15, or— 13 

And x= 3, or — 2f 

4. Reduce 5 a;^ — 20 a; = — 15. Ans. a; = 3, or 1. 

5. Reduce Ta:2 — 8 a; =12f. Ans. a; = f ± f V 26. 

6. Reduce Y a;'' — 4:ax=-—-' Ans. a; =: — , or — -• 

Y. Reducer- ^"7 =x — 3. Ans. x= 10, or 31. 

14 — z S •* 



EQUATIONS OF THE SECOND DEGREE. 185 

8. Reduce ar^ + l^ = — i^ — 4. 

' 4 4 

9. Reduce — = g — = — ^ 



THIRD METHOD OF COMPLETING THE SQUARE. 

178i The method of the preceding Article introduces 
fractions whenever the numerical coefficient of x is not 
even. To present a method of completing the square 
without introducing any fraction, we will again reduce 
equation (1) in Art. 174. 



1. Reduce ax^ -{- dx = e. 



OPERATION. 




aar^-j- dx = e 


(1) 


' a a 


(2) 


^■+"a'*'"*"4a«""4a«"+"a 


(3) 


a»a:» + 4arfx + rf2==ef- + 4ae 


W 


2ax + d= ± Vrf^ + 4«e 


(5) 


_ — d± v/d»4-4atf 


(6) 



Dividing (1) by a, the coefficient of 2*, we have (2); then com- 
pleting the square according to the Rule in Art. 1 75, we have (3) ; 
and if we multiply (3) hy 4 a\ it will give (4), an equation free 
from fractions (unless o, d, or c in (1) are themselves fractions), 
and one whose first member is the square of a binomial. To pro- 
duce this equation directly from (1), we have only to multiply (1) 
by 4a; i. e. by four times the coefficient of z*, and add to both 
members rf* ; i. e. the square of the coefficient of x. Reducing we 
have (S), which is a general expression for the value of x in any 
equation in the form o{ ax^ -{- dx = e. 



186 ELEMENTARY ALGEBRA. 

Hence, to reduce an affected quadratic equation, we 
have this third 

RULE. 

JReduce the equation to the form ax^ -\- dx=z e; then mul- 
tiply the equation by four times the coefficient of x^ and add 
to each member the square of the coefficient of x. 

Extract the square root of each member, and then reduce as 
in simple equations. 

Note. — The third Note under the Rule in Art 177 is applica- 
ble in all cases. 

2. Reduce 5x'^ — 1x = 24:, 

OPERATION. 

5 a:2 — 7 r = 24 
Multiplying by 5 X 4 and 

adding 7^ to each member, 100.r — { ) + 49 = 49 + 480 = 529 
Extracting the square root, lOx — 7 = ±23 

Transposing, 10 x = 7 ± 23 = 30, or — 16 

Whence, a: = 3, or — 1.6 

Note. — The multiplication of the coefficient of x^ need only be 
expressed. Its coefficient after evolving is double its original coef- 
ficient. 

3. Reduce ^^"^"^^^ = 293. 

OPERATION. 
44 x^ — 15 a: 



= 293 



7 

Clearing of fractions, 44 a;^ — 15 a: = 2051 

Completing square, 176 X 44ar2 — { ) + 225 = 225 + 360976 = 361201 

Evolving, 88 a: — 15 = ± 601 

Transposing, 88 a: = 15 ± 601 = 616, or — 586 

293 

Whence, a: = 7, or -— 

44 

4. Reduce 1 x^ — 15 a; = — 2. Ans. a: = 2, or |. 



EQUATIONS OF THE SECOND DEGREE. 187 

6. Reduce —5 ^ — rir = ^ — 3. 

XT O X -f- V 

6. Reduce — 1—^ + - == 5. 

7. Reduce — ^ + ? = 3. 

o „j 4X-I-4 3x— 3 

8. Reduce ' -z 

X 2x — 1 

1 3 

9. Reduce 



7 — 2x ' 2x-f-4 



Ans. x = 


1, or- 


-28. 


Ans. X = 


3, or — 


■u- 


Ans. X = 


= 2, or - 


-i 


lOx-flO 






3x 






13 
10* 







10. Reduce ^x^ — a^ = x — b. 



h , lAcf — V 
Ans. a: = -±y/-3^^. 

11. Reduce 5 — 3x-^ = llOar-*. 
Note. — Multiply by x*. 

12. Reduce s/l^ + V^* = 6 \/^. 
Note. — Divide by y^x! 

179. The rules which have been given for the solution 
of affected quadratic equations apply equally well to any 
equation containing but two powers of the unknown quan- 
tity whenever the index of one power is exactly twice that 
of the other. By the same reasoning as in Art. 1T4, it can 
be shown that all such equations can be reduced to the 
fonn 

aa:^" -|" ^^ = ^> 
or 

It will be seen that the first member is composed of 
two terms so related that they may be the first two terms 
of a binomial square, and we can supply the third by one 
of the rules already given for completing the square. 



OPERATION. 




x^ — 2x^ = 4.8 


(1) 


r6_.2x«+ 1 = 1 + 48 = 49 


(2) 


x^—l=±1 


(3) 


x^ = S,or — 6 


(4) 


a: = 2, or ^— 6 


(5) 



188 ELEMENTARY ALGEBRA. 

1. Reduce ;r^ — 2a:^==48. 

Since the squu*e root 
of x^ isa^, it is evident 
that the second term 
contains as one of its 
factors the square root 
of the first term ; i. e. 
the first member of the 
equation is composed 
of two terms so related that they may be the first two terms of the 
square of a binomial. Completing the square, we have (2) ; extract- 
ing the square root of each member of (2), we obtain (3) ; transpos- 
ing we have (4), and extracting the cube root of (4) we have x = 2, 
or ^^^ 

2. Reduce Sx^ — 4.x^ = 160. 

OPERATION. 

3a;l_4^l— 160 (1) 

36;k^— + 16 = 16+ 1920 = 1936 (2) 

6a;^ — 4=±44 (3) 

6 a:^ = 48, or — 40 (4) 

x^ = 8, or — 5f- (5) 



. x^'= 2, or ^ — -s^o (6) 

a:= 16, or (— Y)^ (7) 

In this equation the index of the higher power is exactly twice 
that of the lower. Completing the square we have (2) ; extract- 
ing the square root of each member of (2), we have (3) ; trans- 
posing, we have (4), which divided by 6 gives (5); extracting the 
cube root of (5), we have (6), which involved to the fourth power 
gives (7). 

3. Reduce ^ + ^ = A. Ans. x = ^J, or — ^J. 

it 4 o 2i 

4. Reduce -^a;^ + f/^x=:l. Ans. a; = j, or — 8. 



EQUATIONS OF THE SECOND DEGREE. 189 

5. Reduce a: — § >/ x = 44^. Ans. x = 49, or 40|. 

6. Reduce x* — x^ = 0. Ans. a; = 1, or 0. 

7. Reduce 3ar* — 2ar»+3 = 228. 

Ans. X =z ± S, or ± 5 \/ — ^. 

8. Reduce 3 x^- — 2 x" = 8. 



Ans. x = aC^2, or <^ — J. 
9. Reduce ^A^-±J = t±ll . Ans. x = 64, or 4. 

10. Reduce x* — a x^ = i. 



Ans.x = ±y/(|±^^ + J). 



180. A polynomial may take the place of the unknown 
quantity in an affected quadratic equation. In this case 
the equation can be reduced by considering the polyno- 
mial as a single quantity. 

1. Reduce (x — 4)^ — 2 (x — 4) = 8. 

OPERATION. 

(x — 4)2 — 2 (x — 4) = 8 (1) 

(x-4)2-() + l = l-f 8 = 9 (2) 

X — 4--l=±3 (3) 

X = 6 ± 3 = 8, or 2 (4) 

Considering x — 4 as a single term, and completing the square, 
we have (2) ; extracting the square root, transposing, &c., we have 
(4), or X =« 8, or 2. 

Note. — We might put (x — 4)=y; then (x — 4)* = 3^, and 
the equation becomes y* — 2 y = 8. After finding the value of y 
in this equation, x — 4 must be substituted for y. 



2. Reduce V5 + x + /^5 + x = 6. 

Ans. x= 11, or 7ft, 



3. Reduce 4 + 2x — x3 + j^>v/44- 2x — x» = i. 

Ans. X = 1 ± ^\/T9, or 3, or — 1. 



190 ELEMENTARY ALGEBRA. 



4. Reduce x + 7 — T s/ x -f- Y = 8 — hs/x + t. 

Ans. a; = 9, or — 3. 



6. Reduce ix — 6)^ — Z s/ x — 5 = . 

Ans. itr = 9, or 5 + ^25. 



6. Reduce 7?-\-Zx-\- s/ x^ _[_ 3 a: -f- 6 = 14. 
Note. — Add 6 to both members. 

Ans. a; = 2, or — 5, or — f ±1 V^. 



1, Reduce 4 + aj2 — 2a: — 2 V6 — 2a: + x^=l. 



Ans. a: = 3, or — 1, or 1 ± 2 \/ — 1. 

60 



8. Reduce Vx^ + a: + 6 = , — 4. 

Ans. a: = 5, or — 6, or — \±.\ \/377. 

181. Of the methods given for completing the square, 
the first is the best when the coefiScient of the less power 
of the unknown quantity is even, and the coefficient of the 
higher power is unity, or when these become so by reduc- 
tion ; the second method is better than the third when- 
ever the coefficient of the less power of the unknown quan- 
tity is even. When the equation cannot be reduced by 
the first method without introducing fractions, if the co- 
efficient of the higher power of the unknown quantity 
is a perfect square, and the coefficient of the less power 
is divisible without remainder by twice the square root 
of the coefficient of the higher power, the method given 
in Note 2, Art. 177, is the best. Let each of the follow- 
ing equations be reduced by the method best adapted 
to it. 

1. Reduce 4x2— 14 _ 3^:2 __ i2a; — 1. 

Ans. a: = 1, or — 13. 

2. Reduce 36ar^ + 24 a; =1020. 

Ans. a: = 6, or — 5|. 



EQUATIONS OF THE SECOND DEGREE. 191 

3. Reduce x — i + ? = 21. Ans. x = 21, or f . 

„ , 9, — x x — 2 2a:— 11 

4. Reduce —2 6~ = x-S * 

5. Reduce g "~ 5 "" To ~ ^^* 

6. Reduce — ^ — = ~^ — ^* 

Ans. a; = 3(c — d), or 3rf. 

7. Reduce |±^ = ^ + 2§. 

8. Reduce — = 9 — 

X — 4 ^ 

9. Reduce - — y = ^ 

10. Reduce ^ + ^ = ?• Ans. x == 1 ± Vl— «"• 

11. Reduce 3x + 3 = 13 + -• 

3a: — 3 « , 3x — 6 

12. Reduce 5 X — — ^==2xH ^ 

13. Reduce ^-1^ + 1 = ^- 

U. Reduce 2 >v/^ — ^x — *l = 5. 

Ans. X = 16, or 7^. 

15. Reduce 2/s/«^^+3V2^= -^j^=- 

Ans. X = 9 a, or — a. 
15 



16. Reduce 4>\/^ — \/2x+ 1 = ,^^ ■ 

Ans. X == 4, or — 2f . 

17. Reduce 5 V25 — x = 6 \/26 — x -f x — 13. 

Ans. X = 16, or 9. 



192 ELEMENTARY ALGEBRA. 

18. Reduce ^ii = V 4~+ V 2^H^^'. 

Ans. X = 12, or 4. 

19. Reduce 6 + 4x-i — 12 a:-^ = 100 ar-^. 

20. Reduce i ^~^' + 6 .^x — y = y • 

21. Reduce 3 a;* — 24 a;^ — 80 = 304. 



22. Reduce ^ -J- 10 = 1 + 4:r». 



Ans. a: = ± 4, or ± 2 V— 2. 
Ans. a: =: \/"9, or .^. 



23. Reduce 5 x* — 3 a:^ + ?1 = 27, 

' 4 



Ans. a;=^ ± iv 6, or ± 3/v/— ^V- 

24. Reduce 2 x^ — 5 a:^ + 4 = 2. 

25. Reduce 6 a;^ + 1184 = 5a;i 



26. Reduce \/^ + 3 — ^x -f 3 = 2. 

Ans. X = 13, or — 2. 



27. Reduce x^ — x/ar^ -j- a; — 5 = 25 — a:. 

Note. — By transposing — x and subtracting 5 from each mem. 
ber, make the expression without the radical in the first member like 
that under the radical; then complete the square, &c. 



28. Reduce a:^ — 2ar + 3V2a:2 — 6ar~ll=x + 33. 

Ans. a; = 6, or — 3, or f ± ^ \/^T3. 

29. Reduce 21 a:* + 11 a:^ _ 69 =:= 321 — (11 ar* + 5 a;^). 

Ans. x= ± ^ VT3, or ± ^ \/ — lb. 

30. Reduce 7— J—— = ^ 4- 



(2 a: — 4/ 8 ' (2a: — 4)* 

31. Reduce (a:2— 4a:)2z=12x — 3a:2. 

32. Reduce a; + (a;2 — ar)2 = a:2 -I- 6112. 



EQUATIONS OF THE SECOND DEGREE. 193 

PROBLEMS 

PRODUCING AFFECTED QUADRATIC EQUATIONS WITH 
BUT ONE UNKNOWN QUANTITY. 

182. Though the numerical negative values obtained in 
solving the following Problems satisfy the equations formed 
in accordance with the given conditions, they are prac- 
tically inadmissible, and are therefore not given in the 
answers. 

1. Divide 40 into two parts such that the sum of their 
squares shall be 1042. 

SOLUTION. 

Let X = one part ; 

then 40 — x = other part 

Then, x« -}- (40 — x)' = 1042 

Expanding, a^ _|- 1600 — 80 a: -j- x» = 1042 

Transposing and uniting, a:" — 40a: = — 279 

Whence, x = 20 ± 1 1 = 81, or 9 

And, 40 — X = 9, or 31 

2. Divide 20 into two parts such that their product 
will be 99J. Ans. ^ and 10^. 

3. The ages of two brothers are such that the age of 
the elder plus the square root of the age of the younger 
is 22 years, and the sum of their ages is 34 years. What 
is the age of each? • Ans. Elder, 18; younger, 16. 

Note. — The other answers found by reducing the equation, viz. 
25 and 9, satisfy the conditions of the equation only upon consid- 
ering y' 9 = — 3. To make the problem correspond to these an- 
swers, the word "plug** must be changed to "minus." 

4. A merchant had two pieces of cloth measuring to- 
gether 96 yards. The square of the number of yards in the 



194 ELEMENTARY ALGEBRA. 

longer is equal to one hundred times the number of yards 
in the shorter. How many yards are there in each piece ? 

Ans. 60 and 36. 

5. Find two numbers whose difference is 3, and the 
sum of whose squares is 117. Ans. 9 and 6. 

6. A merchant having sold a piece of cloth that cost 
him S42, found that if the price for which he sold it were 
multiplied by his loss, the product would be equal to the 
cube of the loss. What was his loss ? 

Note. — If the word " loss " were changed to gain; the other an- 
swer, — 7, or as it would then become, -}- 7> would be correct. 

Ans. $6. 

T. Find two numbers whose difference is 5, and prod- 
uct 176. Ans. 11 and 16. 

8. There is a square piece of land whose perimeter in 
rods is 96 less than the number of square rods in the 
field. What is the length of one side ? Ans. 12 rods. 

9. Find two numbers whose sum is 8, and the sum of 
whose cubes is 152. 

10. A man bought a number of sheep for $240, and 
sold them again for $6.75 apiece, gaining by the bargain 
as much as 5 sheep cost him. How many sheep did he 
buy ? Ans. 40. 

11. Find two numbers whose difference is 4, and the 
sum of whose fourth powers is 1312. 

Note. — Let x — 2 and x -|- 2 be the numbers. 

Ans. 2 and 6. 

12. A man sold a horse for $312.50, and gained one 
tenth as much per cent as the horse cost him. How 
much did the horse cost him? Ans. $250. 

13. The difference of two numbers is 5, and the less 
minus the square root of the greater is 7. What are the 
numbers? Ans. 11 and 16. 



EQUATIONS OF THE SECOND DEGREE. 195 

14. A and B started together for a place 300 miles dis- 
tant. A arrived at the place 7 hours and 30 minutes be- 
fore B, who travelled 2 miles less per hour than A. How 
many miles did each travel per hour? 

Ans. A, 10 ; B, 8 miles. 

15. A gentleman distributed among some boys $15; 
if he had commenced by giving each 10 cents more, 5 
of the boys would have received nothing. How many 
boys were there? Ans. 30. 

16. Find two numbers whose sum is a, and product h. 

Ans. "f-^ and ^-^^ 

17. A merchant bought a piece of cloth for $45, and 
sold it for 15 cents more per yard than he paid. Though 
he gave away 5 yards, he gained $4.50 on the piece. 
How many yards did he buy, and at what price per yard ? 

Ans. 60 yards, at 75 cents per j'^ard. 

18. A certain number consists of two figures whose 
sum is 12; and the product of the two figures plus 16 is 
equal to the number expressed by the figures in inverse 
order. What is the number? Ans. 84. 

19. From a cask containing 60 gallons of pure wine a 
man drew enough to fill a small keg, and then put into 
the cask the same quantity of water. Afterward he drew 
from the cask enough to fill the same keg, and then there 
were 41 j gallons of pure wine in the cask. How much 
did the keg hold ? Ans. 10 gallons. 

20. There is a rectangular piece of land 75 rods long 
and 65 rods wide, and just within the boundaries there is 
a ditch of uniform breadth running entirely round the 
land. The land within the ditch contains 29 acres and 
96 square rods. What is the width of the ditch ? 

Ans. .5 of a rod. 



196 ELEMENTARY ALGEBRA. 



SECTION XX. 

QUADRATIC EQUATIONS CONTAINING TWO 
UNKNOWN QUANTITIES. 

183. The Degree of any equation is shown by the sum 
of the indices of the unknown quantities in that term in 
which this sum is the greatest. Thus, 

^xy — 2a: = 7 is an equation of the second degree, 
hx'y'-^- xif^a'c " '' " fourth 

5^4 _ 14^ _ ^2^.3 .. Ci i^ fifth 

Note. — Before deciding what degree an equation is, it must be 
cleared of fractions, if the unknown quantities appear both in the 
denominators and in the numerators or integral terms; and also 
from negative and fractional exponents. 

184. A Homogeneous Equation is one in which the sum 
of the exponents of the unknown quantities in each term 
containing unknown quantities is the same. Thus, 

^x^ — ^xy -\- y"^ z=:i\^ 
or x^-\- 3:r^2 + 3ar2y + / = 2T 

or x^ — ^.x^yAr Gx^/ — 4a;/ + ^^ = 256 

is a homogeneous equation. 

185. Two quantities enter Symmetrically into an equa- 
tion when, whatever their values, they can exchange places 
without destroying the equation. Thus, 

x^ — 2xy + / — 25 
or a:»+3ar2y + 3:r/4- /= 8 

or x' + 2xy-\-y''^ 2x + 2y = 24 



QUADRATIC EQUATIONS. 197 

186. Quadratic equations containing two unknown quan- 
tities can generally be solved by the rules already given, 
if they come under one of the three following cases: — 

I. When one of the equations is simple and the other 
quadratic. 

II. When the unknown quantities enter symmetrically 
into each equation. 

111. When each equation is quadratic and homogeneous. 

CASE I. 

187. When one of the equations is simple and the other 
quadratic. 

1. Given 1^^+^^,"= ^- I, to find X and y. 



OPERATION. 

2x-f2y = 22 (1) 3x'-{-if=in (2) 

3/=ll— x(3) 3z»-f-121 — 22a:-|-3:»=lll (4) 

42^— 22x = — 10 (5) 

4«a:«—()-fll«= 121— 40 = 81 (6) 

4x= 11 ±9 = 20, or 2 (7) 

y=6,orl0^(9) a:=5, or^ (8) 

From (1) we obtain (3), or y = 11 — x. Substituting this value 
of y in (2), we obtain (4), an affected quadratic equation, which 
reduced gives (8) ; and substituting these values of x in (3), we 
obtain (9). 

In this Case the values of the unknown -quantities can 
generally be found by substituting in the quadratio eqvaHon 
the value of one unknown quantity found by reducing the 
siviple equation. 



198 ELEMENTARY ALGEBRA. 

2. Given | ^^ " ^^ | , to find a: and y. 

xy = 2^ (1) 



OPERATION, 






X — y = 3 




(2) 


x' — 2xy + 7f= 9 




(3) 


4:xy ==112 




(^) 


x^ + 2xy + y'=l2l 


(S) 


x + y =±n 




(6) 


2x = 14, or — 


8 


a) 


2y = 8, or — 


14 


(8) 


X = 1, or — 


4 


(9) 


y = 4, or — 


7 


(10) 



Subtracting four times (1) from the square of (2), we obtain (5) ; 
extracting the square root of each member of (5), we obtain (6); 
adding (2) to (6), we obtain (7) ; subtracting (2) from (6), we ob- 
tain (8) ; and reducing (7) and (8), we obtain (9) and (10). 

Note. — Though Example 2 can be solved by the same method 
as Example 1, the method given is preferable. 

By this method find the values of x and y in the fol- 
lowing equations : — 

3. Given |^ "^ = H. Ans. j ^ = ^• 

4. Given |^+^=13|. Ads. j^ = ^^or6. 

6. Given < ^ ^ >■ • 

\5x + y = 29> 

6. Given j ^y-24> 

iSx — 2yz=lQi 





Zxy 


= 45 




^y 


= 15 


7?- 


-2xy+y« 


= 4 




^ — y 


= ±2 



QUADRATIC EQUATIONS. 199 

CASE II. 
188. When tlie unknown quantities enter symmetrically 
into each equation. 

1. Given | ^ + ^ = ^ J | , to find x and y. 
(x'-j-y*= 152) 

OPEIIATION. 

x+y = % (I) 2:»+y=152 (2) 

a:^ + 2x^ + 3/^ = 64 (3) 

ar«- xy + y^=19 (4) 

(5) 

(6) 

a) 

(8) 

2x = 10, or 6 (9) 
2y = 6, or 10 (10) 

X = 5, or 3 (11) 

y = 3, or 5 (12) 

Squaring (1), we obtain (3); dividing (2) by (1), we obtain (4); 
subtracting (4) from (3), we obtain (5), from which we obtain (6) ; 
subtracting (6) from (4), we obtain (7) ; extracting the square root 
of each member of (7), we obtain (8) ; adding (8) to (1), wc ob- 
tain (9); subtracting (8) from (1), we obtain (10); and reducing 
(9) and (10), we obtain (11) and (12). 

Note 1. — It must not be inferred that x and y are equal to 
each other in these equations ; for when x = 5, y = 3 ; and when 
X = 3, y = 5. In all the equations under this Case the values of 
the two unknown quantities are interchangeable. 

Note 2. — Although 3* -^ r^ = \b2 is not a quadratic equation, 
yet as we can combine the two given equations in such a manner 
as to produce at once a quadratic equation, we introduce it here. 



200 ELEMENTARY ALGEBRA. 

2. Given i„.„ ^^ r»to find x and y. 

U2-fy2_2a; — 2y = 3> ^ 

OPERATION. 

a:y=6 (1) a:2_|_^_2a;_2y= 3 (2) 

2a:y =12 (3) 

(^ + 2//-2(a: + 2/) = 15 (4) 

(3; + yy- () + 1 = 16 (5) 

x + 3/=l±4 = 5,or — 3 (6) 

a: = 3,or2,or ""^^/~^\ 7) 



y=2,or3,or^:iiHyj:iL^(8) 

Adding twice (1) to (2), we obtain (4) ; completing tlie square 
in (4), we obtain (5) ; extracting the square root of each member 
of (5), and transposing, we obtain (6) ; and combining (1) and (6) 
as the sum and product are combined in the preceding example, 
we obtain (7) and (8). 

In Case II. the process varies as the given equations 
vary. In general the equations are reduced by a proper 
combination of the sum of the squares, or the square of the 
sum or of the difference, with multiples of the product of 
the two unknoiun quantities; and finally, of the sum, with 
the difference of the two unknown quantities. 

Note 3. — When the unknown quantities enter into each equation 
symmetrically in all respects except their signs, the equations can be 
reduced by this same method ; e. g. ar — ?/ = 7, and x^ — t/' = 511. 
In such equations the values of the unknown quantities are not inter- 
changeable. 

Note 4. — The signs ± ^ standing before any quantity taken in- 
dependently are equivalent to each other ; but when one of two quan- 
tities is equal to ± a while the other is equal to ^ &, the meaning is 
that the first is equal to -f- a, when the second is equal to — b; and 
the first to — a, when the second is equal to -|- ^« 



QUADRATIC EQUATIONS. 201 

By this method find the values of x and y in the follow- 
ing equations : — 

3. Given i2-+2y= IH . Ans. j- = 4'Or3. 

l3x»+3/ = 273) (y = 3,or4. 

(^^y— 8) .^„ (x = 9,or— 1. 

4. Given ^ . •^, w«« f • -^^s* "^ , o 

(ar» — / = 728) Cy= 1, or — 9. 

6. Given K + ^^t ^'^ «tl ' 
Note. — Divide the second equation by the first. 
6. Given |^-^^ + y = U. 

CASE III. 
189. When each equation is quadratic and homogeneous. 

1. Given pa:y+ y^ = 5) to find a: and y. 
KZt? — ary = 10 ) 

OPERATION. 

2xy+y» = 5 (1) Sa:'* — xy=10 (2) 

Let x=zvy 
2vy^J^f = b (3) 3t;2y2 — vy2=io (4) 



5 10 


a) 


2i;-^-l ~ Zv'—v 


. 15t;» — 5v=20v+ 10 


(8) 


3r^ — 6t; = 2 


(9) 


r = 2, or — ^ 


(10) 



y=±l,or±Vl5 (12) 
x = ry= ±2, or qp i\/T5 (13) 



202 ELEMENTARY ALGEBRA. 

Substituting vy for x in (1) and (2), we obtain (3) and (4) ; from 
(3) and (4) we obtain (5) and (6) ; putting these two values of y^ 
equal to each other, we obtain (7), which reduced gives (10) ; sub- 
stituting this value of v in (5), we obtain (11), which reduced gives 
(12) ; and substituting hx x = vy the values of v and y from (10) and 
(12), we obtain (13). 

Examples under Case III. can generally be reduced best 
by substituting for one of the unknown quantities ih^ product 
of the other by some unknown quantity, and then finding the 
value of this third unknown quantity. When the value of 
this third quantity becomes known, the values of the given 
unknown quantities can be readily found by substitution. 

Note. — Whenever, as in the example above, the square root is 
taken twice, each unknown quantity has four values ; but these values 
must be taken in the same order, i. e. in the example above, when 
y = -j-l, x = -f~2; when y = — l,a: = — 2; when y = -[- y^Ts, 
X •«= — \ y/Ts ; and when y = — y/ 15, a; = -]- ^ V^ 15. 

By this method find the values of x and y in the follow- 
ing equations : — 



2. Given j ^^ - ^^ = 1^ 



a:= ± T, or ± 4\/— -J. 
y= ±5, or dbllV— ~i. 



Ans. < 
3. Given \-'' + ^-y = ^n. 

'•1 



Ans. ^-=±3,or±9^/-i. 
= ± 2,.or TS-i/— i- 



ixy — ^y = 3 — x^ ) 

Ans j^=±2,or±24V-V^. 
\y=± l,or qillV— T^T- 

5. Given j'' ": '^^ = '/ ^^^l • 
( x2 — 3=y2 + 2 > 



QUADRATIC EQUATIONS. 203 

190« Find the values of x and y in the following 

Examples. 
Note. — Some of the examples given below belong at the same 
time to two Cases. Thus in Example 1 both the equations are 
symmetrical, and both are (juadratic and homogeneous, and there- 
fore it belongs both to Case II. and Case III. Example 3 belongs 
both to Case I. and Case II. 



1. Given I , -y = 20) ^^^ Cx=±6,or±4. 



a; = 7, or 3. 
y = 3, or 7. 



ar» + ^-^ = 41 ) (y = ± 4, or ± 5. 

2. Given I ^y= 6 ) 

U^+7x?/ = 55— y2) 

3. Given i^+y= "^ I. 

Ans. j^ = 4^^^8- 
(y = 3, or 4. 

4. Given | ^^=12 > 

(a:» + x = 32— y— y^> 

5. Given \ ^T^'^^^l- Ans. | 

(sxy — 7 = 56) 

6. Given \ ^-y= H. 

(a^y''-\-2xyz= 1295) 

Note. — Considering xy a single quantity, find its value in the 
second equation. 

7. Given i^y--f=m. 

Note. — Subtract from the second equation three times the first, 
and extract the cube root of each member of the resulting equation. 

8. Given j2x=y + 2x/= 168) 

-Ans. 1^ = 1'°' !• 
- (« =3, or4. 



204 



9. Given 



10. Given 



11. Given 



12. Given 



13. Given 



ELEMENTARY ALGEBRA. 
\ a:y=10 j 



na;2 — 2a:y=:88) 



Ans. |^ = 



= ±5. 

±2. 



Ans. 



{ 



a:=±4, or± 66 a^ 
y= ±3, or q: Its V^. 



FJrg-- 



{ 



2a: + 2y = 30~y2. 
4xy = 60 > 



(6a:^-— 23/^= 10 ) 



Ans. 



fa-rr: 1000, or 8. 



14. Given | 3a:^:^18) 
(0:^ — 2/^ = 65) 

Ans. ■! 



625, or 1 



a:= ± 3, or ± 2/^— 1, 
3^=: ± 2, or ± 3^"^=^. 



16. Given J^ (o:-^/) == 3 (V^^ + Vy) ) 



{ 



a;y = 36 



Ans. - 



±\/- 


-23- 


-11 


Tv/- 


2 
-23- 


-11 



, or 9, or 4. 
, or 4, or 9. 



16. Given 



IT. Given 



I x + y = Al ) 

Jo:*— /= n 

(x — 2^ =19) 



QUADRATIC EQUATIONS. 205 

18. Given -{^ ^ ^ :f l . 

i X + y = 65 > 



19. Given i^-'-^^^i^^ 



20. Given i-* + 2x^y + 2xy^ + / = 95| 

(x«— x'f/— xy^ + y»= 5> 

21. Given ■[ ^y= H. 

U* + y* = 272) 

22. Given j^ +.y = H- 

U* + y* = 626) 



PROBLEMS 

PRODUCING QUADRATIC EQUATIONS CONTAINING TWO 
UNKNOWN QUANTITIES. 

191. Though the numerical negative values obtained 
in solving the following Problems satisfy the equations 
formed in accordance with the given conditions, they are 
practically inadmissible, and, except in Example 4, are 
not given in the answers. 

1. The sum of the squares of two numbers plus the 
sum of the two numbers is 98 ; and the product of the 
two numbers is 42. What are the numbers ? 

Ans, 7 and 6. 

2. If a certain number is divided by the product of its 
figures the quotient will be 3 ; and if 18 is added to the 
number, the order of the figures will be inverted. What 
is the number "^ Ans. 24. 

3. A certain number consists of two figures whose 
X roduct is 21 ; and if 22 is subtracted from the number. 



206 ELEMENTARY ALGEBRA. 

and the sum of the squares of its figures added to the 
remainder, the order of the figures will be inverted. What 
is the number? Ans. 37. 

4. Find two numbers such that their sum, their prod- 
uct, and the difierence of their squares shall be equal to 
one another. Ans. f ± ^ V5 and ^ ± ^ V5. 

6. There are two pieces of cloth of difterent lengths; 
and the sum of the squares of the number of yards in 
each is 145 ; and one half the product of their lengths 
plus the square of the length of the shorter is 100. What 
is the length of each ? 

Ans. Shorter, 8 ; longer, 9 yards. 

6. Find two numbers such that the greater shall be to 
the less as the less is to 2f, and the difference of their 
squares shall be 33. 

7. The area of a rectangular field is 1575 square rods ; 
and if the length and breadth were each lessened 5 rods, 
its area would be 1200 square rods. What are the length 
and breadth ? 

8. Find two numbers such that their sum shall be to 
6 as 9 is to the greater, and the sum of their squares 
shall be 45. Ans. 9 Vl and 3 VT, or 6 and 3. 

9. The fore wheels of a carriage make 2 revolutions 
more than the hind wheels in going 90 yards ; but if the 
circumference of each wheel is increased 3 feet, the car- 
riage must pass over 132 yards in order that the fore 
wheels may make 2 revolutions more than the hind wheels. 
What is the circumference of each wheel ? 

Ans. Fore wheels, 13^ feet ; hind wheels, 15 feet. 

10. Find two numbers such that five times the square 
of the greater plus three times their product shall be 104, 
and three times the square of the less minus their prod- 
uct shall be 4. 



KATio AXD r::o?onTio-M. 207 

SECTION XXI. 

RATIO AND PROPORTION. 

102. Ratio is the relation of one quantity to another of 
the same kind ; or, it is the quotient which arises from 
dividing one quantity by another of the same kind. 

Ratio is indicated by writing the two quantities after 
one another with two dots between, or by expressing the 
division in the form of a fraction. Thus, the ratio of a to 
b is written, a : h, or ^ ; read, a is to b, or a divided by b, 

193. The Terms of a ratio are the quantities compared, 
whether simple or compound. 

The first term of a ratio is called the antecedent, and the 
other the consequent; and the two terms together are called 
a covplet. 

194. An Inverse, or Reciprocal Ratio, of any two quan- 
tities is the ratio of their reciprocals. Thus, the direct ratio 

of a to ft is a : b, i. e. r; and the inverse ratio of a to ft is 

1 1 . 1 1 ft , 

- : Tf 1. e. - -T- 7 = -» or ft : a. 
a b aba 

195. Proportion is an equality of ratios. Four quan- 
tities are proportional when the ratio of the first to the 
second is equal to the ratio of the third to the fourth. 

The equality of two ratios is indicated by the sign 
of equality (==) or by four dots (: :). 

Thus, a : b = c : d, or a : b: : c : d, or -r=z-,; read, a to ft 

h a 

equals c to d, or a is to ft as c is to d, or a divided by ft 
equals c divided by d. 



208 ELEMENTARY ALGEBRA. 

196i In a proportion the antecedents and consequents 
of the two ratios are respectively the antecedents and con- 
sequents of the proportion. The first and fourth terms are 
called the extremes, and the second and third the means. 

197. When three quantities are in proportion, e. g. 
a : b = b : c, the second is called a mean proportional be- 
tween the other two ; and the third, a third proportional 
to the first and second. 

198. A proportion is transformed by Alternation when 
antecedent is compared with antecedent, and consequent 
with consequent. 

199. A proportion is transformed by Inversion when 
the antecedents are made consequents, and the conse- 
quents antecedents. 

200. A proportion is transformed by Composition when 
in each couplet the sum of the antecedent and consequent 
is compared with the antecedent or with the consequent. 

201. A proportion is transformed by Division when in 
each couplet the diff'erence of the antecedent and conse- 
quent is compared with the antecedent or with the con- 
sequent. 

THEOREM I. 

202. In a proportion the product of the extremes is equal 
to the product of the means. 



Let a : b = c : d 

i. e. 

Clearing of fractions, 



a c 

h~d 



RATIO AND PROPORTION. 209 

THEOREM II. 
203t If the product of Ivjo quantities is equal to the prod- 
uct of two others, the factors of either product may he made 
the extremes^ and the factors of the other the means of a 
proportion. 

Let ad=:bc 

Dividing by hd, h^^d 

i. e. a : b = c : d 

THEOREM III. 

204. If four quantities are in proportion, they will he in 
proportion by alternation. 

Let a : b = c : d 

By Theorem I. ad=bc 

By Theorem II. a -. c=^b : d 

THEOREM IV. 

205. If four quantities are in proportion, they mil he in 
proportion by inversion. 

Let a : b = c : d 

By Theorem I. ad= be 

By Theorem 11. b : a = d : c 

THEOREM V. 

206. If three quantities are in proportion, the product of 
tJie extremes is equal to the square of tJie mean. 

Let a : b = b : c 

By Theorem I. ac = If^ 

THEOREM VI. 

207. If four quantities are in proportion, they will he in 
proportion hy composition. 



a 
b — 


c 
'd 




t + ^ = 


a ' 




b ~ 


c + rf 
d 




a + b: b = 


c + d: 


d 



210 ELEMENTARY ALGEBRA. 

Let a \h:=.c \ d 

\. e. 

Adding 1 to each member; 

or 

i. e. 

THEOREM VII. 

208. If four quantities are in proportion, they will he in 
proportion by division. 

Let a : b =z c : d 

i. e. 

Subtracting I from each member 

or 

i. e. a 

THEOREM VIII. 

209. Two ratios respectively equal to a third are equal 
to each other. 

Let a : b=:m '. n and c : d=m : n 

. ^ a m . c m 

1. e. - = - and ,== - 

on d n 

Hence (Art. 13, Ax. 8), |= f 

i. e. a '. b = c : d 

THEOREM IX. 

210. If four quantities are in proportion, the sum and 
difference of the terms of each couplet will he in proportion. 



a c 
b~~d 




l-^-l-^ 




a — b c — d 
b ~ d 




-b: b = c — d: 


d 



RATIO AND PROPORTION. 211 

Let a : h:=c : d 

By Theorem VI. a -\- b : b = c -{- d : d {I) 

and by Theorem VII. a^—b . b = c — d : d (2) 

From (1 ), by Theorem III. a-{'b:c-\-d=b:d 

From (2), by Theorem III. a — b:c — dz=b:d 

By Theorem VIII. a + b:c + d=a — b : c — d 

Hence, by Theorem III. a -\- b : a — b = c -\- d : c — d 

TUEOREM X. 
211* Equimultiples of two quantities have tJw same ratio 
as tfie quantities themselves. 

For by Art. 83, ? = ^ 

•^ mo 

i. e. a : b = ma : mb 

Cor. It follows that either couplet of a proportion may 
be multiplied or divided by any quantity, aiid the result- 
ing quantities will be in proportion. And since by Theo- 
rem III. if a : b = ma : mb, a : ma == b : mb, or ma : a 
= mb : b, it follows that both consequents, or both ante- 
cedents, may be multiplied or divided by any quantity, 
and the resulting quantities will be in proportion. 

THEOREM XI 
212, If four quantities are in proportion, like powers or 
like roots of these quantities will he in proportion. 

Let a : b = c : d 

a c 
b=d 

Hence, ^=^ 

i. e. t^ : l^ = c" : d^ 

Since n may be either integral or fractional, the theorem 
is proved. 



212 ELEMENTARY ALGEBRA. 

THEOREM XII. 

213. If any number of quantities are proportional, any 

antecedent is to its consequent as the sum of all the antece- 
dents is to the sum of all the consequents. 

Let a : b = c : d=ze :f 

Now ab=zab (1) 

and by Theorem I. ad=zbc (2) 

and also af^=:be (3) 

Adding(l), (2), (3), a(H-^ + /) = b {a -\- c -^ e) 

Hence, by Theorem II. a : b =z a-\- c -\-e •.b-\- d -\-f 

THEOREM XIII. 

214. If there are two sets of quantities in proportion, their 
products, or quotients, term by term, wilt be in proportion. 



Let 


a : b = c : d 




and 


e:f=g :k 




By Theorem I. 


ad = bc 


(1) 


and 


eh=fy 


(2) 


Multiplying (1) by (2), 


adeh = b cfg ; 


(3) 


Dividing (1) by (2), 


ad be 


(4) 


From (3), by Theorem IL 


ae : bf=zcg : dh 




and from (4), 


a b c d 





PROBLEMS IN PROPORTION. 

215. By means of the principles just demonstrated, a 
proportion may often be very much simplified before 
making the product of the means equal to the product 
of the extremes ; and a proportion which could not oth- 
erwise be reduced by the ordinary rules of Algebra may 
often be so simplified as to produce a simple equation. 



RATIO AND PROPORTION. 213 

1. The cube of the smaller of two numbers multiplied 
by four times the greater is 96 ; and the sum of their 
cubes is to the difference of their cubes as 210 : 114. 
What are the numbers ? 

SOLUTION. 

Let X = the greater and y = the less. 

Then 4x/ = 96 (1) x'+y :ar» — / = 210 : 114 (2) 
From (2), by Theo. X^ Cor. ar' + y' : a:* — y* = 35 : 19 
By Theorem IX. 2a:» : 2y» = 64 : 16 

By Theorem X., Cor. a:« : / = 27 : 8 

By Theorem XI. a: : y = 3 : 2 

By Theorem I. 2x = 3y (3) 

From (1) and (3) we find x = 3 and y = 2. 

2. The product of two numbers is 78 ; and the differ- 
ence of their cubes is to the cube of their difference as 
283 : 49. What are the numbers ? 

SOLUTIOX 

I^t X = the greater and y = the less. 

Thenxy=78 (1) I* — 3/»: a:» — 3x«y + 3a:y» — y» = 283 : 49 (2) 
From (2), by division, Sx^y — 3 x^ : (x — y)' = 234 : 49 

Dividing 1st couplet by a: — y, 3xy :(x — y)' = 234 : 49 

Dividing antecedents by 3, xy : (x — y)' = 78 : 49 

Substituting the value of zy, 78 : (x — y)* = 78 : 49 

Dividing antecedents by 78, 1 : (x — y)' = 1 : 49 

Extracting the square root, 1 : x — y = 1 : 7 

Whence, x — y = 7 (3) 

From (1) and (3) we find x = 13 and y = 6. 

3. The sum of the cubes of two numbers is to the cube 
of their sum as 13 : 25 ; and 4 is a mean proportional be- 
tween them. What are the numbers? 



214 ELEiMENTARY ALGEBRA. 

4. The difference of two numbers is 10 ; and their prod- 
uct is to the sum of their squares as 6 : 3*7. AVhat are 
the numbers ? 

SOLUTION. 

Let X = the greater and y = the less. 

Then a;— 2/= 10 (1) xy\ t? -\- f = ^ -. ^1 {^^ 

From (2), by Theorem X., Cor. 2xy\ o? -{- f = \1 -.^1 

By Theorem IX. o? -\- Ixy -{- y^ \ x^ — 'Ixy -\- f = 4.^ -. lb 

By Theorem XL x ■\- y \ x — y = 7:5 

By Theorem IX. ' 2a::2^/=12:2 

By Theorem X., Cor. x\y =6:1 

By Theorem I. a; = 6 y (3) 
From (1) and (3) we find a; = 12 and y = 2. 

5. The product of two numbers is 136 ; and the dif- 
ference of their squares is to the square of their differ- 
ence as 25 : 9. What are the numbers ? Ans. 8 and IT. 

6. As two boys were talking of their ages, they dis- 
covered that the product of the numbers representing 
their ages in years was 320, and the sum of the cubes 
of these same numbers was to the cube of their sum as 
T : 27. What was the age of each? 

Ans. Younger, 16; elder, 20 years. 

Y. As two companies of soldiers were returning from 
the war, it was found that the number in the first multi- 
plied by that in the second was 486, and the sum of the 
squares of their numbers was to the square of the sum as 
13 : 25. How many soldiers were there in each company? 

Ans. In 1st. 27; in 2d, 18. 

8. The difference of two numbers is to the less as 100 
is to the greater ; and the same difference is to the greater 
as 4 is to the less. What are the numbers ? 

J^OTE. — Multiply the two proportions together. (Theorem XIII.) 



PROGRESSION. 215 

SECTION XXII. 

PROGRESSION. 

216. A Progression is a series in which the terms in- 
crease or decrease according to some fixed law. 

217. The Terms of a series are the several quantities, 
whether simple or compound, that form the series. Tlie 
first and last terms are called the extremes, and the others 
the means. 

ARITHMETICAL PROGRESSION. 

218. An Arithmetical Progression is a series in which 
each term, except the first, is derived from the preced- 
ing by the addition of a constant quantity called the com' 
mon difference. 

219. When the common difference is positive, the series 
is called an ascending series, or an ascending progression ; 
when the common difference is negative, a descending se- 
ries. Thus, 

a, a -\- d, a -\- 2d, a -\- Zd, &c. 

is an ascending arithmetical series in which the common 
difference is d] and 

a, a — d, a — 2d, a — Zd, &c. 

is a descending arithmetical series in which the common 
difference is — d. 

220. In Arithmetical Progression there are five elements, 
any three of which being given, the other two can be 
found : — 

1. The first term. 

2. The last terra. 



216 ELEMENTARY ALGEBRA. 

3. The commou difference. 

4. The number of terms. 

5. The sum of all the terms. 

221. Twenty cases may arise in Arithmetical Progres- 
sion. In discussing this subject we shall let 

a = the first terra, 

I = the last term, 

d = the common difference,- 

n = the number of terms, 

S=. the sum of all the terms. 

CASE I. 

222. The first term, common difference, and number of 
terms given, to find the last term. 

In this Case a, d, and n are given, and / is required. The suc- 
cessive terms of the series are 

a, a -\- d, a -\- 2d, a -{- Sd, o -j- 4 rf, &c. ; 

that is, the coefficient of d in each term is one less than the number 
of that term, counting from the left ; therefore the last or nth term in 
the series is 

a-\- (n—1) d 
•or Z = a -J- (n — 1) d 

in which the series is ascending or descending according as d is posi- 
tive or negative. Hence, 

RULE. 

To the first term add tJie product formed by multiplying the 
common difference by the number of terms less one. 

1. Given « = 4, d =z 2, and w = 9, to find /. 

l=:a-\-(n — I) d = 4: + (9 — 1) 2 = 20, Ans. 

2. Given a = t, d=3, and n = 19, to find /. 

Ans. /=61. 



PROGRESSION. 217 

3. Given a = 29, rf = — 2, and n = 14, to find I 

Ans. / = 3. 

4. Given a = 40, d=z 10, and n = 100, to find /. 

5. Given a = 1, d=z ^, and n =. 17, to find /. 

6. Given a = |^, rf = — ^^, and n = 13, to find /. 

7. Given a = .01, d = ^ .001, and n = 10, to find /. 

CASE II. 
223. The extremes and the number of terms given, to 
find the sum of the series. 

In this Case a, /, and n are given, and S is required. 

Now 5 = o + (o + d) + (a + 2d) + (a + 3d) + + Z 

or, inrerting the seriee, 5= i + ( i — d) + {l — 2d) -\- (l — Sd) + -fa 

Adding these together, 2 5= (a + + (a +0 + (o + + (a + + + (a + 

And since (a -j- /) is to be taken as many times as there are terms, 
hence 2 5 = n (a -f- 

or S = - (a -\- I). Hence, 

RULE. 
Find one half (he product of the sum of the extremes and 
the number of term^. 

Note. — If in place of the last term the common difference is 
given, the last term must first be found by the Rule in Case I. 



1. Given a = 3, /= 141, and n = 26, to find S. 

^= ^ (a + = IT (3 + 141) = 1872, Ans. 

2. Given a = |, / = 25, and n = 63, to find S. 

Ans. ^=793f. 

3. Given a = 4, <f = 2, and n = 24, to find S. 

Ans. aS'=648. 

4. Given a = — 3, c? = 2, and n = 4, to find S. 

Ans. 5=0. 
10 



218 ELEMENTARY ALGEBEA. 

5. Given a = ^, c? = — ^, and n-=z 3, to find S. 

6. Given a =z .07, I = .17, and n = 11, to find aS'. 

7. Given «= — 4J-, c?= |, and w = 25, to find aS. 

CASE III. 
224. The extremes and number of terms given, to find 

the common difference. 

In this case a, I, and n are given, and d is required. 
From Case I. we have Z = a -f- (« — 1) <i 
Transposing and reducing, d = . Hence, 

RULE. 
Divide the last term minus the first term by the number 
of terms less one, and the quotient will be the common dif- 
ference. 

1. Given a = 5, I=z4z1, and n == 7, to find d. 

I —a 47 — 5 ^ , 

2. Given a = 27, 1= 148, and n = 12, to find d. 

Ans. d=z 11. 

3. Given a = 41, ^=3, and n = 20, to find d. 

Ans. d = — 2, 

4. Given a = -^l, 1 = ^^, and » = 6, to find d. 

Ans. d = — 2%. 

5. Given a = .09, /= .9, and w = 10, to find d. 

Note. — This rule enables us to insert any number of arithmet- 
ical means between two given quantities ; for the number of terms 
is two greater than the number of means. Hence, if m = the num- 
ber of means, m4-2 = n, or m 4- 1 == n — 1, and d = — p-r* 

m-j- 1 

Having found the common difference, the means are found by add- 
ing the common difference once, twice, &c., to the first term. 



PROGRESSION. 219 

6. Find 6 arithmetical means between 3 and 38. 

Ans. 8, 13, 18, 23, 28, 33. 

T. Find 3 arithmetical means between 3 and 27. 

8. Find 5 arithmetical means between 1 and 3Y. 

9. Find 7 arithmetical means between 2 and 26. 
Note. — When m= 1, the formula becomes 

-'-^ 

Adding a to each member, 

But a -\- d is the second term of a series whose first term is a and 
common difference </, or the arithmetical mean of the series a, a -|- rf, 
a -\- 2d. Hence, the arithmetical mean beticeen two quantities is one 
half of their sum. 

10. Find the arithmetical mean between 7 and 17. 

Ans. 12. 

11. Find the arithmetical mean between ^ and J. 

12. Find the arithmetical mean between 4 and — 4. 

225. From the formulas established in Arts. 222 and 
223, viz. 

lz=a + {n — l)d (1) 

S=l(a + l) (2) 

can be derived formulas for all the Cases in Arithmetical 
Progression. 

From (1) we can obtain the value of any one of the four quanti- 
ties, /, a, n, or d, when the other three are given ; and from (2) 
the value of any one of the four quantities, «S, n, a, or /, when the 
other three are given. Formulas for the remaining twelve Cases 
which may arise are derived by combining the two formulas (1) 
and (2), so as to eliminate that one of the two imknown quantities 
whoM value is not sought 



220 ELEMENTARY ALGEBRA. 

1. Find the formula for the value of n, when a, rf, and 
S are given. 

OPERATION. 



Z =« + (w — l)rf (1) 8z=^ 


n . 


+ 


I) (2) 


ln = an + dn^ — dn (3) 2S—anz= 


-.In 




(4) 


an-{-dn'^ — dn = 2S^q,n 






(5) 


^2 (d-2a\ 2S 






(6) 


"" \ d )''— d 


(d-2ay_(d-2ay 2S 






a) 



?2 



2d 



(8) 



To obtain the formula required in this example, I must be elim- 
inated from (1) and (2). From (1) and (2) we obtain (3) and (4). 
Placing these two values of Zn equal to each other, we form (5), 
which reduced gives (8), or the value of n in known quantities. 

2. Find the formula for the value of n, when d, I, and 

S are given. An« « — 21+ d ±vj2T-^ dy -8dS 

Ans. n — 2^ 

3. Find the formula for the value of *S', when a, d, and 
n are given. Ans. S=in [2 a -)- (w — I) d]. 

4. Find the formula for the values of S, when a, d, and 
/ are given. ^^^ ^ __ (^ + «) (l — a-^d) 

z d 

5. Find the formula for the value of S, when d, n, and 
I are given. Ans. S=^n[2l — (n — 1) d]. 

6. Find the formula for the value of I, when a, d, and 
S are given. * z ^ i fl ^Y 



Ans. «=-i±y/(«-|) +2<^'S- 



T. Find the formula for the value of I, when d, n, and 
S are given. * ^_ 2 S -^n (n — i) d ^ 

2n 



PROGRESSION. 221 

8. Find the formula for the value of d, when a, n, and 

S are given. . , 2 6' — 2 an 

~ n (n — 1) * 

9. Find the formula for the value of d, when. a, I, and 

S are given. ^ ^ ^ ibpll^l. ' 

2 5 — (/ -f- a) 

10. Find the formula for the value of d, when n, I, and 

5 are given. Ans. rf = ?^?^^. 

n (n — 1) 

11. Find the formula for the value of a, when rf, n, and 
5 are given. . 2 5 — n (n — 1) rf 

2n 

12. Find the formula for the value of a, when d, I, and 



S are given. Ans. a = f ± J g + /)' _ 2 rf 5. 

226. To find any one of the five elements when three 
others are given. 

RULE. 
Substitute the given values in that formula whose first mem- 
ber is the required term, and whose second contains the three 
given terms. 

1. Given d = 2, 1=21, and *S'= 120, to find a. 

OPERATION. 



^=l±\j{l+^^y-^'^'i^o (2) 

a = 3, or — 1 (3) 

In Example 12, Art 225, we find (I), the required formula; substi- 
tuting the given values of d, /, and 5, we obtain (2), which reduced 
gives (3), or a = 3, or — 1. 

Note. — If a = 3, n= 10; butifa = — 1, n = 12. 



222 ELEMENTARY ALGEBRA. 

2. Given d=^, 1=21, and *S'=:392, to find n. 

Ans. n = 147, or 16. 
Note. — When n = 147, a = — 21| ; but when n = 16, a = 22. 

3. Given d=:1, n = Q, and S=: 135, to find I 

Ans. 1:= 40. 

4. Given c? = — 2, ti = 6, and a = 5, to find aS'. 

Ans. ,^=0. 

5. Given « = ^, / = 15, and S =z 2881, to find d. 

Ans. c? = f . 

6. Find the 100th term of the series 3, 10, 17, &c. 

Ans. 696. 
T. Find the sum of 100 terms of the series 3, 10, 17, &c. 

Ans. 34950. 

8. Find the common difference and sum of the series 
whose first term is 25, last term 95, and number of 
terms 15. Ans. c?=5; 1^=900. 

9. Find the sum of the natural series of numbers from 
1 to 100, inclusive. 

10. Find the sum of 10 of the odd numbers 1, 3, 5, &c. 

11. Find the sum of 10 of the even numbers 2, 4, 6, &c. 

12. How many strokes does a clock strike in 12 hours? 

13. If 100 trees stand in a straight line 10 feet from 
one another, how far must a person, starting from the 
first tree and returning to it each time, travel to go to 
every tree ? Ans. 18J miles. 

14. If a person should save a cent the first day, two 
cents the second, three the third, and so on, how much 
would he save in 365 days? Ans. $667.95. 

15. If a person should save $25 a year and put this 
sum at simple interest at 5 per cent at the end of each 
year, to how much would it amount at the end of 25 
years ? 



PROGRESSION. 223 

PROBLEMS 
TO WHICH THE FORMULAS DO NOT DIRECTLY APPLY. 

227. Sometiraes in examples in progression the terms 
are not directly given, but are implied in the conditions of 
the problem. In this case the formulas cannot be directly 
used, but the terms can be represented by unknown quan- 
tities, and equations formed according to the given con- 
ditions. 

228. If X = first term and y = the common difference ; 
then 

X, X + y, X -f 2 y, x -f 3 y, &c. 

will represent the series. 

It will often be found more convenient when the num- 
ber of terms is odd to represent the middle term by x 
and the common difference by y ; then the series for three 
terms will be 

X — y, X, X + y ; 
and for five terms, 

X — 2 y, X — y, x, x + y, x -f- 2 y ; 

and when .the number of terms is even, to represent the 
two middle terms by x — y and x -[- y» and the common 
difference by 2y; then the series for four terms is 

X — 3y, X — y, x + y, x + 3y. 

The advantage of this latter method is, that the sum of 
the series, or the sum or difference of an}' two terms 
equally distant from the mean, or means, will contain but 
one unknown quantity. 

1. The sum of three numbers in arithmetical progres- 
sion is 15, and the sum of their squares is 83. What 
are the numbers ? 



224 ELEMENTARY ALGEBRA. 

Let X =: the mean term and y = the common diiference ; 
then the series will be x — y, x, and x -\- y. 
By the conditions, 3 a; =15 (1) 

and Bx'' + 2f = SS (2) 

Ans. 3, 5, 1. 

2. The sum of four numbers in arithmetical progres- 
sion is 44, and the sum of the cubes of the two means 
is 2926. Ans. 5, 9, 13, 17. 

3. Find seven numbers in arithmetical progression such 
that the sum of the first and fifth shall be 10, and the 
difference of the squares of the second and fourth 40. 

4. There are four numbers in arithmetical progression ; 
the product pf the first and third is 20 ; and the product 
of the second and fourth 84. What are the numbers ? 

Ans. 2, 6, 10, 14. 

6. The sum of four numbers in arithmetical progression 
is 32 ; and their product 3465. What are the numbers ? 

Ans. 5, Y, 9, 11. 

6. The sum of the squares of the extremes of four num- 
bers in arithmetical progression is 461 ; and the sum of 
the squares of the means 425. What are the numbers ? 

Ans. 10, 13, 16, 19. 

Y. A certain number consists of three figures which 
are in arithmetical progression ; if the number is divided 
by the sum of its figures, the quotient will be 15 ; and 
if 396 is added to the number, the order of the figures 
will be inverted. What is the number i* Ans. 135. 

8. Find four numbers in arithmetical progression such 
that the sum of the squares of the first and. third shall be 
104, and of the second and fourth 232. 

9. Find four numbers in arithmetical progression such 
that the sum of the squares of the first and second shall 
be 29, and of the third and fourth 185. 



PROGRESSION. 225 

SECTION XXIII. 

GEOMETRICAL PROGRESSION. 

229. A Geometrical Progression is a series in which 
each term, except the first, is derived by multiplying the 
preceding term by a constant quantity called the ratio. 

230* If the first term is positive, when the ratio is a 
positive integral quantity, the series is called an ascending 
series, and when the ratio is a positive proper fraction^ a 
descending series ; but if the first term is negative, the sc- 
ries is ascending when the ratio is a positive proper frac- 
tion, and descending when the ratio is a positive integral 
quantity. Thus, 

2, 6, 18, 54, &c. ) J. 

' ' ' ' > are ascendine: series : 

-^64,-18,— 6,— 2, &c.) ^ 

64, 32, 16, 8, &c.> , ,. 

' ' ' ' V are descending series. 

— 8, — 16, — 32, — 64, &c. > ^ 

If the ratio is negative, the terms of the progression are 
alternately positive and negative. Thus, if the ratio is 
— 2 and the first term 3, the series will be 

3, — 6, + 12, — 24, + 48, &c. ; 
but if the first term is — 3, 

— 3, +6, —12, +24, —48', &c. 

The positive terms of these two series constitute an as- 
cending progression whose ratio is the square of the given 
ratio ; and the negative terms a descending progression 
having the same ratio. 

231* In Geometrical Progression there are five elements, 
any three of which being given, the other two can be found. 
These elements are the same as in Arithmetical Progres- 
sion, except that in place of the common difference we have 
the ratio. 

10* o 



226 ELEMENTARY ALGEBRA. 

232t Twenty cases may arise in Geometrical Progres- 
sion. In discussing these cases we shall preserve the 
same notation as in Arithmetical Progression, except 
that instead of d = the common difference we shall use 
r = the ratio. 

CASE I. 

233. The first term, ratio, and number of terms given, 
to find the last term. 

In this Case a, r, and n are given, and I required. 
The successive terms of the series are 

a, ar, ar^^ ar^, a?-*, &c. 
That is, each term is the product of the first term and that power 
of the ratio which is one less than the number of that term count- 
ing from the left; therefore the last or nth term in the series is 

or l=sar^-^. Hence, 

RULE. 
Multiply the first term by that power of the ratio whose 
index is one less than the number of terms. 

1. Given a = Y, r = 3, and w = 5, to find /. 

/=ar"-^ = 1 X 3* =567, Ans. 

2. Given a = 3, r = 2, and w = 9, to find I. 

Ans. /=768. 

3. Given a = 64, r == J, and n = 10, to find /. 

Ans. I = ^. 

4. Given a = — T, r = — 4, and w = 3, to find /. 

Ans. Z = — 112. 

5. Given a = — ^, r = ^, and w = 5, to find /. 

Ans. 1==. — y¥s- 

6. Given a = 5, r = — \, and n = 10, to find I. 

7. Given a == — ^, r=^, and w = 8, to find /. 

8. Given a = — 10, r = — 2, and n = 6, to find /. 



PROGRESSION. 227 

CASE II. 

234. The extremes, and the ratio given, to find the sum 
of the series. 

In this Case ct, /, and r are given, and S is required. 

Now S == a -{- ar -{- ar^ -\- at^ -{- -f- / (1) 

Multiplying (1) by r, rS=ar-\- aH -f ar* -f -|-^+^'* (2) 

Subtracting (1) from (2), rS — S= Ir — a 
Whence, <?»= . Hence, 

RULE. 
Multiply the last term hy the ratio, from tJie product sub- 
tract the first term, and divide iJie remainder by the ratio 
less one. 

1. Given a = 2, /== 20000, and r = 10, to find S. 

S='^ = 'O^lXJl - ' = 22222, Ans. 

2. Given a = 7, / = 45927, and r = 3, to find S. 

Ans. 5=68887. 

3. Given a = — 5, l = — 405, and r r= 3, to find S. 

4. Given a = — ttVjj ^ = i* ^^^ ^ = — *?» to ^^^ *S'. 

Ans. S=^%o^. 

CASE III. 

235. The first term, ratio, and number of terms given, 
to find the sum of the series. 

In this Case a, r, and n are given, and S required. 
The last term can be found by Case I., and then the sum of the 
series by Case II. Or better, since 

lr = ar* 
Substituting this value of Z r in the formula in Case II. we have 

r" 1 

S a« — — — X a- Hence, 



228 ELEMENTARY ALGEBRA. 

RULE. 
From the ratio raised to a power whose index is equal to 
the number of terms subtract one, divide the remainder by 
the ratio less one, and multiply the quotient by the first term. 

1. Given a =: 4, r =z^, and ?i == 5, to find aS'. 

^= ^^-=4 X « = ^^ X 4 = 11204, Ans. 
r — 1 7 — 1 

2. Given a=.^, r = 6, and n = 6, to find S. 

V Ans. aS'= 558. 

3. Given a =: ^, r = i, and n = Y, to find *S'. 

Ans. S=iU. 

4. Given a=. — 5, r = — 4, and n =: 4, to find S. 

Ans. ^5^=255. 

6. Given a = — |, r = Q, and w = 5, to find *S'. 
6. Given a = §, r = — 3, and w = 6, to find S. 
Y. Given a = — }, r = 2, and n = 8, to find aS'. 



». In a geometrical series whose ratio is a proper frac- 
tion the greater the number of terms, the less, numeri- 
cally, the last term. If the number of terms is infinite, the 
last terra must be infinitesimal ; and in finding the sum 
of such a series the last term may be considered as noth- 
ing. Therefore, when the number of terms is infinite, the 
formula 

S =z becomes 



Sz= 



r — 1 1 — r 



Hence, tg find the sum of a geometrical series whose ra- 
tio is a proper fraction and number of terms infinite, 

RULE. 
Divide the first term by one minus the ratio. 



PROGRESSION. 229 

1. Find the sum of the series 1, J, i, &c. to infinity. 

•5=f^ = r^ = 2, Ans. 

2. Find the sum of the series f, |, y^* ^c. to infinity. 

Ans. ^jj. 

3. Find the sum of the series -, ^, -j, &c. to infinity. 

Ans. r- 

c — 1 

4. Find the sum of the series 6, 4, 2 J,' &c. to infinity. 

Ans. 18. 
6. Find the value of the decimal .4444, &c. to infinity. 

Note. — This decimal can be written ^ -J- ^^ -f~ njW» ^* 

Ans. |. 

6. Find the value of .324324, &c. to infinity. 

7. Find the value of .32143214, &c. to infinity. 

CASE IV. 
237. The extremes- and number of terms given, to find 
the ratio. 

In this Case a, /, and n are ^ven, and r is required. 
From Case I. l=:at*-^ 

Whence, r= 4/-. Hence, 

RULE. 
Divide the last term hy the firstj and extract that root of 
the quotient whose index is one less than the number of 
terms. 

1. Given a = 7, I = 567, and n = 5, to find r. 

,-1/7 



■^i = ^iiI==3.A„s. 



2. Given a = 6f , / = i, and n = 6, to find r. 

Ans. r =z }. 



230 ELEMENTARY ALGEBRA. 

3. Given a = — I, 1= 31^, and n = 4:, to find r, 

Ans. r =. — 5. 

Note. — This rule enables us to insert any number of geometri- 
cal means between two numbers ; for the number of terms is two 
greater than the number of means. Hence, if in == the number of 

means, m-(-2 = n, orm-}-l == n — 1; and r = t /-• Having 

found the ratio, the means are found by multiplying the first term 
by the ratio, by its square, its cube, &c. 

4. Find three geometrical means between 2 and 512. 

Ans. 8, 32, 128. 

5. Find four geometrical means between 3 and 3072. 

Ans. 12, 48, 192, 768. 

6. Find three geometrical means between 1 and y'g^. 

Ans. ^, i, i. 

Note. — When m = 1 , the formula becomes 



= \/^ 



Multiplying by a, ar = a^/-=i/aL 



sjl-^'- 



But ar is the second term of a series whose first term is a and 
ratio r ; or the geometrical mean of the series a, ar, a 1^. ' Hence, 
the geometrical mean between two quantities is the square root of their 
product. 

7. Find the geometrical mean between 8 and 18. 

Ans. 12. 

8. Find the geometrical mean between ^ and 343. 

Ans. 7. 

9. Find the geometrical mean between \ and i-^2b' 

10. Find the geometrical mean between — J and — s-tVt- 



PROGRESSION. 231 

238. From the formulas established in Arts. 233 and 234, 
l=ar^-^ (1) 

5='rl" • (2) 

can be derived formulas for all the Cases in Geometrical 
Progression. 

From (1) we can obtain the value of any one of the four terms, /, 
a, n, or r, when the other three are given; from (2), the value of 5, 
/, r, or a, when the other three are given. Formu\^s for the remaining 
twelve Cases which may arise are derived by combining the formulas 
(1) and (2) so as to eliminate that one of the two unknown terms 
whose value is not sought. 

1. Find the formula for the value of *S', when /, n, and 
r are given. 

From (1), ,3^11 = « 

Substituting this value of a in (2), Sz 



or • ' S = 



r— 1 
/(r--l) 



(r_i)r«-i 

Note. — The four formulas for the value of n cannot be derived or 
used without a knowledge of logarithms ; and four others, when n ex- 
ceeds 2, cannot be reduced without a knowledge of equations that can- 
not be reduced by any rules given in this book. 

239. To find any one of the five elements when three 
others are given. 

RULE. 

Subsliiufe in that one of the formulas (1) or (2) that con- 
tains the four elements, viz. the three given and the one re- 
quired, the given values, and reduce the resulting equation. 

If neither formula contains the four elements, derive a for- 
mula that will contain them, then substitute and reduce the 
resulting equation ; or substitute the given values before deriv- 
ing the formula, then eliminate tJie superfluous element and 
reduce the resulting equation. 



232 ELEMENTARY ALGEBRA. 



1. Given r = 3, w = 


5, 


and 


[ S=126, to find /. 




/=a/^-i (1) 






^ Ir — a 


(2) 


l=Sla (3) 






V26 = ^^-« 


<4) 


8l = « (') 






a = 3/— U52 


(6) 


3 Z— 1452 




I 

81 


(7) 




242? 




1452 X 81 (8) 





? = 486 (9) 

Substituting the given values of r, n, and 5 in (1) and (2), we 
obtain (3) and (4) ; finding the value of a, the superfluous element, 
from (3) and (4), and putting these values equal to each other, we 
form (7), an equation containing but one unknown quantity. Re- 
ducing (7) we obtain (9), or I = 486. 

2. Given a == 4, r = b, and ASf= 15624, to find I. 

Ans. 1= 12500. 

3. Given a = 2, n = 5, and /= 512, to find S. 

Ans. 5=682. 

4. Find the formula for the value of a, when r, n, and 

S are given. . (r — l) 'S' 

Ans. a — ^ _— • 

5. A gentleman purchased a house, agreeing to pay one 
dollar if there was but one window, two dollars if there 
were two windows, four if there were three, and so on, 
doubling the price for every window. There were 14 
windows. How much must he pay? Ans. $8192. 

6. A man found that a grain of wheat that he had sown 
had produced 10 grains. Now if he sows the 10 grains 
the next year, and continues each year to sow all that is 
produced, and it increases each year in tenfold ratio, how 
many grains will there be in the seventh harvest, and how 
many in all ? j^ ( In Yth harvest, 10000000 grains. 

(In all, lUlllU grains. 



PROGRESSION. 233 

PROBLEMS 
TO WHICH THE FORMULAS DO NOT DIRECTLY APPLY. 

240. In solving Problems in Geometrical Progression, 
if we let X ■=. the first term and y ^ the ratio, the series 
will be 

a:,-ary, xf, xf, &c. 

It will often be found more convenient to represent the 
series in one of the following methods : — 

1st When the number of terms is odd, 
a^» ^^!h y 



for three terms; 

3» 7/* 

-, I*, xy, y*, ~ for five terms. 

2d. When the number of terms is even, 

a" «» 

-, X, y, — for four terms; 

y a; 

X* X" y" t^ r. ' 

-0 1 -» a:, y, -, -^ for six terms. 
f y -^ x' x* 

Which method is most convenient in any case will de- 
pend upon the conditions that are given in the problem. 

1. There are three numbers in geometrical progression, 
the greatest of which exceeds the least by 32 ; and the 
difierence of the squares of the greatest and least is to 
the sum of the squares of the three as 80 : 91. What are 
the numbers ? 

SOLUTION. 

Let X, xy, and xy* represent the scries. Then 

xy«~x = 32(l) x«y*— x»:x» + x«y«-f-x»y« = 80:91 (2) 

y«— l:l-|-y»4-y*==80:91 (3) 

9ly* — 91 = 80-f-80y»-|-80y* (4) 

lly« — 80y»=171 (5) 

x=4 (7) y=3 (6) 



234 ELEMENTARY ALGEBRA. 

Dividing the first couplet of (2) by x^ we obtain (3) ; from (3) 
vre form (4), which reduced gives (G), or y = 3. Substituting the 
value of 1/ in (i), we obtain (7)^ or a; = 4. 

Ans. 4, 12, 36. 

2. The sum of three numbers in geometrical progres- 
sion is 39, and the sum of their squares 819. What are 
the numbers ? 

SOLUTION. 

Let X, s/ xy, and y represent the series. Then 

a: + /v/^y + yr=39 (1) a;^ + xy + ^^ ^ 819 (2) 

ar~>v/^7 + y=r21 (3) 

2x^2y — m (4) 

x- \- y = 3 (5) 

2\/^=18 (6) 

xy = ^\ (1) 

Dividing (2) by (1), we obtain (3); adding (3) to (1), we ob- 
tain (4), which reduced gives (5) ; subtracting (3) from (1), we 
obtain (6), which reduced gives (7). Combining (5) and (7) as 
the sum and product are combined in Example 1, Art. 188, we 
obtain x = 27 and y = 3. 

Ans. 3, 9, 21. 

3. Of four numbers in geometrical progression the dif- 
ference between the fourth and second is 60 ; and the sum 
of the extremes is to the sum of the means as 13 : 4. 
What are the numbers ? 

SOLUTION. 

Let X, xy, xy^, and xy^ represent the series. Then 
xf 



64a: 



xy=:60 (1) 


x/ + x:xy' + xy=l3:i 


(2) 




if-!/ + l :.y=:13:4 


(3) 




4:f — iy + 4,=z 13y 


(i) 


4a: = 60 (7) 


if—11y^ — i 


(5) 


x=l (8) 


y = 4 


(6) 



PROGRESSION. 235 

Dividing the first couplet of (2) by xy -\- x,yre obtain (3) ; from 
(3) we form (4), which reduced gives (G), or y = 4. Substituting 
this value of y in (1) and reducing, we obtain (8), or ar== 1. 

Ans. 1, 4, 16, 64. 

4. Of four numbers in geometrical progression the sum 
of the first two is 10 and of the last two 160. What are 
the numbers ? Ans. 2, 8, 32, 128. 

5. A man paid a debt of S310 at three payments. The 
several amounts paid formed a geometrical series, and the 
last payment exceeded the first by $240. Wliat were the 
several payments? Ans. $10, $50, $250. 

6. In the series x, \/xi/, and y what is the ratio? 

A„s./f. 

7. In the series -, x, y, and - what is the ratio? 

y' ' ^' X 

8. There are four numbers in geometrical progression 
whose continued product is 64 ; and the sum of the series 
is to the sum of the means as 5 : 2. What are the num- 
bers ? Ans. 1, 2, 4, 8. 

9. There are five numbers in geometrical progression ; 
the sum of the first four is 156, and the sum of the last 
four 780. What are the numbers ? 

10. There are three numbers in geometrical progression 
whose sum is 126; and the sum of the extremes is to the 
mean as 17 : 4. What are the numbers? 

11. The sum of the squares of three numbers in geo- 
metrical progression is 2275 ; and the sum of the ex- 
tremes is 35 more than the mean. What are the numbers ? 

12. Of four numbers in geometrical progression the sum 
of the first and third is 52 ; and the difference of the means 
is to the difference of the extremes as 5 : 31. What are the 
numbers ? 



236 ELEMENTARY ALGEBRA. 

SECTION XXIV. 

MISCELLANEOUS EXAMPLES. 

1. From 6ac— 5ab-{-c^ take Sac— [Sab — (c — c^) 
+ 1c}. Ans. Sac — 2ab + 2c^-^6c. 

2. Reduce x^ y^ — ( — xi/'^-{-x^ ) xi/ — x"^ f — {3/^ 

— y {^y — ^^) } ) t^ ^^s simplest form. 

Ans. 2x^y^ -\- x^. 

3. Reduce {a — h -\- cf — fa (c — a — h) -^ [b {a + h 

-f- c) — c (a — b — ^) I ) to its simplest form. 

Ans. 2(a'-\-b'-{-c^). 

4. Reduce (x -\- a) a -{- 1/ — { & + ^) {'^ + ^) — y 
(x -\- a — 1) — {^ -\- y) {b — ct)} to its simplest form. 

Ans. a^ — b"^. 

5. Reduce (a^ — b^)c — {a — b) [a {b^ c) — b {a — c)\ 
to its simplest form. Ans. 0. 

6. Reduce {a -\- b) x — {b — c) c — ^{b — x)b—{b — c) 
{b -\- c)\ — ax to -its simplest form. Ans. 2b x — be. 

1. Multiply a^ + 2aH — SaP by — {—SaH-{- a^ b^). 

8. Multiply a"^ -|- 6 a^ _f- 9 by a^ — 6 a^ + 9. 

9. Multiply a -\- b — c by n — b -\- c. 

10. Divide 2% a- — 6 a^ — Ga^ — 4a^ — 96 a + 264 by 
3a2 — 4a + 11. 

11. Divide 1 — 18x2+810:^ by 1 -\-Qx + ^x\ 

12 Divide ^ a^ -\- I —Aa^ — Q a hy 1 -|- 2 «2 — 3 a. 
13. Divide 9 ^^ — 7 x^/ -\-2y^ by 3 x^ + 2 ^^^ — /. 



MISCELLANEOUS EXAMPLES. 287 

14. Divide 23 a — 30 — T a» + 6 a< by 3 a — 2 a^ — 5. 

15. Find the prime factors of a* — b*. 

16. Find the prime factors of 4m'n'^ — 49 m^n^". 

17. Find the prime factors of x^ — 2xi/'\-y^, 

18. Find the prime factors of x^ — y". 

19. Find the greatest common divisor of 6 a:* — lOx^y 
+ ISy* and 4x« + 8 x'^y + 8 xy^ _|_ 4y8. Ans. x + y. 

20. Find the greatest common divisor of 8 a 6* + 24 a 6* 
+ 16 a 6 and 7 i« + 7 i« + 7 6* — 7 6^ Ans. ^ + b. 

21. Find the greatest common divisor of 6 a:^ -|- ^ ^y — 
Bt/^ and 12x2 _^ 22 ary + 6y^ 

22. Find the greatest common divisor of 4x -|- 4ar'' — 40 
and 3a:r*y — 48 y. Ans. x — 2. 

23. Reduce 7 — i,x / « . » ^ . mv to its lowest terms. 

(a — 6) (a* 4- 2 a 6 -f- ^) 

24. Reduce , ,. to its lowest terms. 

a' — IT 

25. Reduce ., . .^ ^ ~ \, ; — ir to its lowest terms. 

26. Find the least common denominator and reduce 

z r— i ,~i « — -^ 15 to a single fraction. 

1 — a 1 + a 1-f-o 1 — <^ n 

Ans. — 



1 +a« 

27. Find the least common denominator and reduce 

■r-^ — 4 — r— i — i to a single fraction. 

28. Find the least common denominator and reduce 
4a»+3a6 _ 48a'6 ^ • 1 r ^• 
4a«-3a6-^- 16a^-9a«y *^ ^ «^°^^^ ^'^^*^^'^- 

29. Reduce to one fraction with the least possible de- 

. . a y — a*-|-a6 3ft — a , c 

nominator j- j—f U — . 

.bed cd ' bd 



238 ELEMENTARY ALGEBRA. 

30. Reduce to one fraction with the least possible de- 

a + & h 4- c , a -\- c 

nominator yr ^ r — 7 X7 h\ 1 



(b — c){c-'a) {a — c){a — h) ' (6 — a)(c — 6) 

Ans. j-j r-7 r-7 77 = 0. 

(h — c) (c — a) (a — h) 



4- 2 a: /2 — Zx (\Q — x)x\^ . , „ ^. 



31. Find the least common denominator and reduce 
3j-2a: 
Y . . 

Ans. z~. 

2 -\- X 

32. Reduce to one fraction with the least possible de- 
. ^ l+rc Ax \—x . 2x-\-Q7? 

nommator ^j^, - ^^^^ - j^^j^,- Ans. -^^--^, • 

33. Reduce a — c ~ ^^ ^ to its simplest form. 

34. Reduce ^~^ to its simplest form. 

35. Reduce — —- \- x and — —, 2y each to a 

X — 2?/ ' X -\-2y ^ 

single fraction and find their product. Ans. , „ » 

36. Subtract ~^— from 



2^ — a; a;-h?/ 

37. Subtract 3a:-[-^ from a; — 



&« 



88. Multiply (-i^,y by ^-3! 

39. Divide — — by f , , and multiply the result by a'. 

40. Divide ^--^^--^, by -^— ^. 

41. Divide T by (— ri; + z r)' 

m -f- n '' \a-\- a — 0/ 



iUSCELLANEOUS EXAMPLES. 239 

Je^ rk- -J 4(a» — aft) , Sab 

42. Divide -^, — f— ,;,- by -5 rj- 

b{a -\- by "^ a} — b* 

43. Divide --^ i — by — ■ j — , and give 

a — X ' a 4- X '' a — x a -+- x ° 

0*4- z" 
the answer in its lowest terms. Aus. —^ — • 

2ax 

44. Reduce r-r = 1 • What is the value of 

a a -{- b a — o 

X, if a = — 2 and 6 = 3? 

3x — 1 



45. Reduce x 



1 + 



46. 'Reduce(a-{-x){b — x) — a{b — c)-~^-^ = 0. 

What is the value of a:, if a = 2, b = — 3, and c = — 1 . 

47. Reduce -^^^ = — ~ What is the value of 

b a 

ar, if a = 2, 6 = — 1, and c = 3 ? 

48. Reduce a — J-±^ = 0. 

1 — x 

ab 



49. Find the value of x in the equation x = ^^r| ^^rj 

in its simplest form. a — b a-t-b 

60. A man spends $2. He then borrows as much money 
as he has left, and again spends S2. Then borrowing 
again as much money as he has left, he again spends $2, 
and then has nothing left. How much money did he have 
at first? 

61. If 5 is subtracted from a certain number, two thirds 
of the remainder will be 40. What is the number ? 

62. Having a certain sum of money in my pocket, I 
lost c dollars, and then spent one ath part of what re- 
mained and had loft one />th part of what I had at first. 
What was the original sum ? What does the answer be- 
come if a = 3, 6 = 9, and c = 5 ? 



240 ELEMENTARY ALGEBRA. 

63. If I buy a certain number of pounds of beef at $0.25 
a pound, I shall have $0.25 left; but if I buy the same 
number of pounds of lard at $0.15 a pound, I shall have 
$1.25 left. How much money have I? 

54. Divide 84 into three parts so that one third of the 
first, one fourth of the second, and one fifth of the third 
shall be equal. 

65. In a certain orchard 25 more than one fourth of 
the trees are apple trees, 2 less than one fifth are pear 
trees, and the rest, one sixth of the whole, are peach 
trees. How many trees are there in the orchard ? 

56. A merchant spent each year for three years one 
third of the stock which he had at the beginning of the 
year ; during the first year he gained $ 600, the second 
$500, and the third $400. At the end of the three 
years he had but two thirds of his original stock. What 
was his original stock ? 

57. From a cask of wine out of which a third part had 
leaked, 84 liters were drawn, and then the cask was half 
full. What is the capacity of the cask ? 

58. A gentleman has two horses and a chaise. The 
chaise is worth a dollars more than the first horse and h 
dollars more than the second. Three fifths of the value of 
the first horse subtracted from the value of the chaise is 
the same as seven thirds of the value of the second horse 
subtracted from twice the value of the chaise. What is 
the value of the chaise and of each horse ? What are the 
answers if a = — 50 and J = 50 ? 

59. A had twice as much money as B. A gained $30 
and B lost $40. Then A gave B three tenths as much as 
B had left, and had left himself 20 per cent more than he 
had at first. How much did each have at first? 



MISCELLANEOUS EXAMPLES. 241 

60. A number of men had done one third of a piece of 
work in 9 days, when 18 men were added and the work 
completed in 12 days. What was the original number of 
men? 

61. A boatman can row down the middle of a river 14 
miles in 2 hours and 20 minutes ; but though he keeps 
near the shore where the current is one half as swift 
as in the middle, it takes him 4 hours and 40 minutes to 
row back. What is the velocity of the water in the 
middle of the river ? Ans. 2 miles an hour. 

62. A had three fifths as much money as B. A paid 
away $80 more than one third of his, and B $50 less 
than four ninths of his, when A had left one third as much 
as B. What sura had each at first? 

63. A farmer hired a man and his son for 20 days, 
agreeing to pay the man $3.50 a day and the son $1.25 
for every day the son worked ; but if the son was idle, 
the farmer was to receive $0.50 a day for the son's board. 
For the 20 days' labor the man received $67. How 
many days did the son work ? 

64. I purchased a square piece of land and a lot of three- 
inch pickets to fence it. I found that if I placed the pick- 
ets 3 inches apart, I should have 50 pickets left ; but if I 
placed the pickets 2^ inches apart, I must purchase 60 
more. How much land and how many pickets did I pur- 
chase ? Ans. 18906J square feet and 1150 pickets. 

65. A criminal having escaped from prison travelled 10 
hours before his escape was discovered. He was then 
pursued and gained upon 3 miles an hour. When his 
pursuers had been on the way 8 hours, they met an ex- 
pressman going at the same rate as themselves, who had 
met the criminal 2 hours and 24 minutes before. In what 
time from the commencement of the pursuit will the crimi- 
nal be overtaken ? Ans. 20 hours. 

11 F 



242 ELEMENTARY ALGEBRA. 

66. In February, 1868, a man being asked the time, 
answered that the number of hours before the close of the 
month was exactly one sixth of 10 less than the number 
that had passed in the month. What was the exact 
time ? Ans. February 25th, 10 o'clock, p. m. 

6*?. A and B owned adjoining lots of land whose areas 
were as 3 : 4. A sold to B 100 hectares of his, and after- 
ward purchased of B two fifths of B's entire lot ; and then 
the original ratio of their quantities of land had been re- 
versed. How much land did each own at first ? 

Ans. A, 300 ; B, 400 hectares. 

68. A laborer Was hired for TO days ; for each day he 
wrought he was to receive $2.25, and for each day he was 
idle he was to forfeit $0.75. At the end of the time he 
received $118.50. How many days did he work? 

69. A sum of money was divided equally among a num- 
ber of persons by giving to the first $100 and one sixth of 
the remainder, then to the second $200 and one sixth of 
the remainder, then to the third $300 and one sixth of the 
remainder ; and so on. What was the sum divided and 
wjiat the number of persons ? 

70. A besieged garrison had a quantity of bread which 
would last 9 days if each man received two hectograms a 
day. At the end of the first day 800 men were lost in a 
sally, and it was found that each man could receive 2f 
hectograms a day for the remainder of the time. What 
was the original number of men ? 

71. Find a fraction such that if 1 is added to the de- 
nominator its value will be ^ ; but if the denominator is 
divided by 3 and the numerator diminished by 3, its value 
will be f . 

72. If 7 years are added to A's age, he will be twice as 
old as B ; but if 9 years are subtracted from B's age, he 
will be one third as old as A. What is the age of each ? 



MISCELLANEOUS EXAMPLES. 213 

Y3. A, B, and C compare their fortunes. A says to 
B, " Give me S700 of your money, and I shall have twice 
as much as you retain." B says to C, "Give me $1400, 
and I shall have three times as much as you retain." 
C says to A, "Give me S420, and I shall have five times 
as much as you retain." How much has each? 

74. An artillery regiment had 39 soldiers to every 5 
guns, and 4 over, and the whole number of soldiers and 
oflScers was six times the number of guns and officers. 
But after a battle in which the disabled were one half of 
those left fit for duty, there lacked 4 of being 22 men to 
every 4 guns. How many guns, how many officers, and 
how many soldiers were there ? 

Ans. 120 guns, 44 officers, 940 soldiers. 

75. A car containing 5 more cows than oxen was started 
from Springfield to Boston. The freight for 4 oxen was 
$2 more than the freight for 5 cows, and the freight for 
the whole would have amounted to $30 ; but at the end 
of half the journey 2 more oxen and 3 more cows were 
taken into the car, in consequence of which the freight of 
the whole was increased in the proportion of 6 to 6. What 
was the original number of cows and oxen, and what was 
the freight for each ? 

. j 9 cows and 4 oxen. 

I Freight for a cow, $2 ; for an ox, $3. 

76. Two sums of money amounting together to $1600 
were put at interest, the less sum at 2 per cent more than 
the other. If the interest of the greater sum had been in- 
creased 1 per cent, and the less diminished 1 per cent, the 
interest of the whole would have been increased one fif- 
teenth ; but if the interest of the greater had been increased 
1 per cent while the interest of the other remained the 
same, the interest of the whole would have been increased 
one tenth. What were the sums, and the rates of interest? 

Ans. $1200 at 7 per cent ; $400 at 9 per cent. 



244 ELEMENTARY ALGEBRA. 

TT. A and B can perform a piece of work together in 
lYf days. They work together 10 days, and. then B fin- 
ishes the work alone in 16| days. How long would it 
take each to do the work ? 

T8. The Emancipation Proclamation of President Lincoln 
was promulgated on the 1st day of January in a year rep- 
resented by a number that has the following properties: 
the second (hundred's) figure is equal to the sum of the 
third and fourth minus the first ; or to twice the sum of 
the jfirst and fourth ; the third is a third part of tlie sum 
of the four; and if 1818 is added to the number, the order 
of the figures will be inverted. What was the year ? 

79. A and B can do a piece of work in a days ; A and 
C in b days ; B and C in c days. In how many days can 
each do it ? 

80. A can do a piece of work in a days, B in ^ days, 
and C in c days. In how many days can A and B together 
do it? B and C together? A and C together? All three 
together ? 

81. A market-man bought some eggs for $0.28 a dozen, 
and sold some of them at 3 for 8 cents and some at 5 for 12 
cents, receiving for the whole $6.24, and clearing $0,64. 
How many did he sell at each rate ? 

82. One cask contains 56 liters of wine and 40 of water, 
and another 96 of wine and 16 of water. How many liters 
taken from each cask will make a mixture containing 52 
liters of wine and 24 of watei- ? 

83. A and B are travelling on roads which cross each 
other. When B is at the point of crossing, A has 120 me- 
ter's to go before he arrives at this point, and in 4 minutes 
they are equally distant from this point ; and in 32 minutes 
more they are again equally distant from it. What is the 
rate of each? Ans. A's, 100; B's, 80 meters a minute. 



MISCELLANEOUS EXAJIPLES. 245 

84. Multiply tT by a:". 

85. Multiply x' by x"'. 

86. Divide y^* by y-^ 

87. IMvide a""* by 0*+*. 

88. Transfer the denominator of — ^5 to the numerator. 

89. Free -zr-f-^ ^^^^^ negative exponents. 

90. Expand (— 2a«)*. 

91. Expand (a^i)"*. 

92. Expand (— 3a:-V)*- 

93. Expand (x'" X a:")'. 

94. Expand {x — >v/y)'. 
96. Expand (a» — 2 6)». 

96. Expand {1x—ff. 

97. Expand (2x — 3)*. 

98. Expand (3 a — 2 6)". 

99. Find five terms of (x — y)*^. 

100. Expand (2 — x — y)«. 

101. Expand (3 ■— a — 6 + cf. 

102. Find ^C^"^. 

103. Find y/^;;r^- 

104. Find V— 16a:«. 

105. Find the square root of | — (2a + 2x — 4j i/^ 

+ a«__4a + 2ax + 4 — 4a: + a:«. 

106. Reduce </ 256 a* 6» — 768 a* 6' c^ to its simplest form 



246 ELEMENTARY ALGEBRA. 

101. Reduce ^ — ^nd tV | to equivalent radicals hav- 
ing a common index. 

108. Add VA. V^> and Vif. 

109. From a^IM take a^JO. 

110. Multiply i^f by i>?^J. 

111. Divide —Sa/TO by V'5. 

112. Divide a^'6 by /</'6. 

113. Find the cube of S a/2x. 

114. Find the square root of 6 -^"3^ 

115. Multiply 6 + \/3 by 3 — a^~3. 

116. Expand (x^ — 2Ai/lcy. 

111. Expand (^^-s/-^y. 

118. Expand (^ I -ly. 

119. The area of a rectangular field is 4 acres and 35 
square rods ; and the sum of its length and breadth is 
equal to twice their difference. What are the length and 
breadth ? 

120. Two travellers, A and B, set out to meet each 
other. They started at the same time and travelled on the 
direct road toward each other. On meeting it appeared 
that A had travelled 18 miles more than B, and that A 
could have travelled B's distance in 9 days, while it would 
have taken B 16 days to travel A's distance. How far 
did each travel ? Ans. A, 12 miles ; B, 54 miles. 

121. Find three quantities such that the product of the 
first and second is a ; of the second and third, b ; and of 
the first and third, c 



MISCELLANEOUS EXAMPLES. 247 

122. A and B invest in stocks. At the end of the 
year A sells his stocks for S108, gaining as much per 
cent as B invested; B sold his for $49 more than he 
paid, gaining one fourth as much per cent as A. What 
sum did each invest? Ans. A, $45; B, $140. 



123. 


Reduce ISar* — 33ar — 40 = 0. 




124. 


r. J X x-fl 1 

Reduce ^_^ T ^e* 




125. 


Reduce (^-y) (- !/)=,,- 






Ans. y = — 2.1 ±3 V 1.49, or 6, or - 


-f 


126. 


Reduce (x* — x" + 4:y + x" = 5T04 + x\ 






Ans. x=±3, or ±2^—2, or ii/^^^v^" 

91 


^^ 


l^T 


RpfliiPA A./2 r -J- 1 _L 2 ^/a: — 










128. 


Reduce y* — 2 A/f — 3y + 5 = 3y — 2. 




129. 


Reduce > ■ » = ^^ — 




130. 






131. 
132. 


Given 1 . l'^*, ^ !^ 1 . to find z and y. 





133. Given \'\'^^~\l},to&ndx and y. 

134. Given } IT^y "^^^V, to find x and y. 

C 5xy=T5) 



248 ELEMENTARY ALGEBRA. 

135. Given i ^ "^^ 7 ^! "^ !!H > to S^d x and y. 

l2x^i/ — 2xf= 140) ^ 

136. Given j^x^y - 2:.^^ = 875 ) ^^ ^^^ ^ ^^^ 

13Y. Given j^''^' "" ^^^, == ^H , to find x and y. 
( a;2 4- y2 __ 25 i ^ 

138. Given j ^ ^' 7 f ^^ = ^^^ I , to find a: and y. 

(3/_j-5a;2/z= 132) ^ 

139. Given 1^' + 2^' + ^ + ^ == ^^l , to find a; andy. 

C X -\- 1/ — xy= 0) 

Ans. 1^=2, orH-3±V2T). 
(y = 2, or i(— 3TV21). 

140. Given 1^^ + ^ = "^U , to find x and y. 

(a:* + 3^* = 1921) ^ 

141. A drover sold a number of sheep that cost him 
$297 for $1 each, gaining $3 more than 36 sheep cost 
him. How many sheep did he sell ? 

142. A merchant sold a piece of cloth for $75, gaining 
as much per cent as the piece cost him. What did it 
cost him? 

143. A drover bought 12 oxen and 20 cows for $920, 
buying one ox more for $160 than cows for $6Q. What 
did he pay a head for each? 

144. A started from C towards D and travelled 4 miles 
an hour. After A had been on the road 6^ hours, B 
started from D towards C, and travelled every hour one 
fourteenth of the whole distance, and after he had been 
on the road as many hours as he travelled miles an hour, 
he met A. What was the distance from C to D ? 



MISCELLANEOUS EXAMPLES. 249 

145. A person bought a number of horses for S1404. 
If there had been 3 less, each would have cost him $39 
more. What was the number of horses and the cost of 
each ? 

146. Find a number of four figures which increase 
from left to right by a common difference 2, while the 
product of these figures is 384. Ans. 2468. 

147. A rectangular garden 24 rods in length and 16 in 
breadth is surrounded by a walk of uniform breadth which 
contains 3996 square feet. What is the breadth of the 
walk? Ans. 3 feet. 

148. A square field containing 144 ares has just within 
its borders a ditch of uniform breadth running entirely 
round the field and covering 381.44 centares of the area. 
What is the breadth of the ditch ? Ans. 0.8 meter. 

149. A and B hired a pasture into which A put 5 horses, 
and B as many as cost him $5.50 a week. If B had put 
in 4 more horses, he ought to have paid $6 a week. What 
was the price of the pasture a week? Ans. $8. 

150. A father dying left $3294 to be divided equally 
among his children. Had there been 3 children less, each 
would have received $ 183 more. How many children 
were there ? 

151. A merchant bought a quantity of tea for $66. If 
he had invested the same sum in coffee at a price $0.77 
less a pound, he would have received 140 pounds more. 
How many pounds of tea did he buy ? 

152. Find two quantities such that their sum, product, 
and the sum of thoir squares shall be equal to one an- 
other. Ans. i (3 ± \/'^^^) and i {S T \/^^^). 

163. Find two numbers such that their product shall 
be 6, and the sum of their squares 13. 
U* 



250 ELEMENTARY ALGEBRA. 

154. A and B talking of their ages find that the square 
of A's age plus twice the product of the ages of both is 
3864 ; and four times this product, minus the square of 
B's age, is 3575. What is the age of each? 

Ans. A's, 42 ; B's, 25. 

155. Find two numbers such that five times the square 
of the less minus the square of the greater shall be 20 ; 
and five times their product minus twice the square of the 
greater shall be 25. . 

156. A and B purchased a wood-lot containing 600 
acres, each agreeing to pay $11500. Before paying for 
the lot, A offered to pay $20 an acre more than B, if B 
would consent to a division and give A his choice of situ- 
ation. How many acres should each receive, and at 
what price an acre ? 

Ans. A, 250 acres at $ TO an acre ; B, 350 at $50. 

157. A merchant bought two pieces of cloth for $175. 
For the first piece he paid as many dollars a yard as there 
were yards in both pieces ; for the second, as many dol- 
lars a yard as there were yards in the first more than in 
the second ; and the first piece cost six times as much as 
the second. What was the number of yards in each 
piece? ^ Ans. In 1st, 10 yards ; in 2d, 5. 

158. Two sums of money amounting to $14300 were 
lent at such a rate of interest that the income from each 
was the same. But if the first part had been at the same 
rate as the second, the income from it would have been 
$532.90 ; and if the second part had been at the same 
rate as the first, the income from it would have been 
$490. What was the rate of interest of each? 

Ans. First, 7 per cent ; second, 7fV P^r cent. 

159. Divide 29 into two such parts that their product 
will be to the sum of their squares as 198 : 445. 



MISCELLAKEOUS EXAMPLES. 251 

160. What is the length and breadth of a rectangular 
field whose perimeter is 10 rods greater than a square 
field whose side is 50 rods, while its area is 250 square 
rods less than the area of the square field ? 

Ans. Length, To rods ; breadth, 30. 

161. A rectangular piece of land was sold for $5 for 
every rod in its perimeter. If the same area had been 
in the form of a square, and sold 'in the same way, it 
would have brought S90 less; and a square field of the 
same perimeter would have contained 272^ square rods 
more. What were the length and breadth of the field ? 

Ans. Length, 49 ; breadth, 16 rods. 

162. A starts from Springfield to Boston at the same 
time that B starts from Boston to Springfield. When 
they met, A had travelled 30 miles more than B, having 
gone as far in IJ days as B had during the whole time; 
and at the same rate as before B would reach Springfield 
in 6f days. How far fi-om Boston did they meet? 

Ans. 42 miles. 

163. The product of two numbers is 90 ; and the dif- 
ference of their cubes is to the cube of their difference as 
13 : 3. What are the numbers? 

164. A and B start together from the same place and 
travel in the same direction. A travels the first day 25 
kilometers, the second 22, and so on, travelling each day 
3 kilometers less than on the preceding day, while B 
travels 14^ kilometers each day. In what time will the 
two be together again ? Ans. 8 days. 

165. A starts from a certain point and travels 5 miles 
the first day, 7 the second, and so on, travelling each day 
2 miles more than on the preceding day. B starts from 
the same point 3 days later and follows A at the rate of 
20 miles a day. If they keep on in the same line, when 
will they be together ? Ans. 8 or t days after B Btarts. 



252 ELEMENTARY ALGEBRA. 

166. A gentleman offered his daughter on the day of 
her marriage $1000; or $1 on that day, $2 on the next, 
$3 on the next, and so on, for 60 days. The lady chose 
the first offer. How much did she gain, or lose, by her 
choice ? 

167. The arithmetical mean of two numbers exceeds 
the geometrical mean by 2 ; and their product divided by 
their sum is 3^. What are the numbers ? 

168. A father divided $130 among his four children in 
arithmetical progression. If he had given the eldest $25 
more and the youngest but one $5 less, their shares would 
have been in geometrical progression. What was the share 
of each ? 

169. The sum of the squares plus the product of two 
numbers is 133 ; and twice the arithmetical mean plus 
the geometrical mean is 19. What are the numbers ? 

110. The sum of three numbers in geometrical progres- 
sion is in ; and the difference of the second and third 
minus the difference of the first and second is 36. What 
are the numbers ? 

171. There are four numbers in geometrical progression, 
and the sum of the second and fourth is 60 ; and the sura 
of the extremes is to the sum of the means as 7 : 3. What 
are the numbers? 



Cambridge : Electrotyped and Printed by Welch, Bigelow, & Ca 



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